% The following TeXfile comprises part of

%     MEASURE THEORY, by D.H.Fremlin
%          (University of Essex, Colchester, England;
%                   e-mail  fremdh@essex.ac.uk)

% Please contact me with any difficulties not covered by the remarks
% below.

% The file is set up for processing by plain TeX in the presence of
% files  amstex.tex ,  amssym.tex  and  amsppt.sty  suitably supported by the
% appropriate fonts.   If these are unavailable, a readable version
% can be obtained by removing the comment symbol  %  from the line

%        %   \amstexsupportedfalse

% below.   Some systems do not allow access to files  amstex.tex  and
% amsppt.sty , but instead employ a command "amstex <filename.tex>";
% for such systems it is sometimes right to delete or disable the
% commands

%        \input amstex
%        \documentstyle{amsppt}

% (I have little experience with this and should welcome comments.)

% Various formats of output are obtainable.   The principal variations
% are

%    1. "Standard" output uses the command
%               \smallprint
% and gives output adapted to A4 paper (21 cm x 29.7 cm).

%    2.  Using the command
%               \USsmallprint
% gives output adapted to 8.5" x 11" .

%    3.  Using the command
%               \noproofs
% produces an abridged version, in which proofs are omitted.

%    4.   Using the command
%               \resultsonly
% produces a super-abridged version, omitting comments and exercises
% as well as proofs.

% These commands are grouped together at the end of the introductory
% definitions;  they may be activated by deleting the initial comment
% symbols  % .

% Any diagrams included will be called for by inputting postscript files.
% (You can look for these by hunting for   \picture  in the code below.)
% To get these you will need the relevant  .ps  files with the expected
% names  ( \picture{mt133g}{...}  demands the file  mt133g.ps , for instance).


%(If you can read TeX fluently, you may wish to fast-forward to a line
%beginning   %file starts here   or   \frfilename
%if one of these is present.)

%TeXfiles with this heading are designed for processing with AmSTeX,
%accessed through TeXfiles  amstex.tex (with amssym.tex  and
% amsppt.sty  and possibly  amsppt.sti)  during compilation.   
%If these are unavailable, a readable
%version can (I hope) be produced by deleting the  %  before
% \amstexsupportedfalse  below.

%Some special difficulties arise with the importation of  .ps  files
%in diagrams.   Look for  \picture  below for notes on these.

%The following \count registers are used for special purposes:
%   \count1=\stylenumber  to specify style
%   \count2=\footnotenumber  to count footnotes

%AmSTeX declaration
\newif\ifamstexsupported
     \amstexsupportedtrue
%     \amstexsupportedfalse               %use when AMSTeX unavailable
\newif\ifDHFamstexloaded        %for detection of a non-standard AMSTeX
     \DHFamstexloadedfalse    
\newif\ifDHFamspptloaded        
     \DHFamspptloadedfalse    
\newif\iffourteensavailable
     \fourteensavailabletrue
    \fourteensavailablefalse
\newif\ifathome
     \athometrue     %to cope with a quirk     of my son's Ghostscript
     \athomefalse
\newif\iffirstpage
     \firstpagetrue

%Style assignments
\countdef\stylenumber=1
\countdef\footnotenumber=2
\def\fullsize{\magnification=\magstep1
     \def\pagewidth{390pt}
     \def\pageheight{9truein}
     \hsize\pagewidth
     \vsize\pageheight
     \stylenumber=1}
\def\smallprint{\magnification=\magstep0
     \overfullrule0pt
     \def\pagewidth{468pt}
     \def\pageheight{9truein}
     \hsize\pagewidth
     \vsize\pageheight
     \stylenumber=2}
\def\printersize{\magnification=\magstep1
     \baselineskip24pt
     \overfullrule0pt
     \def\pagewidth{330pt}
     \def\pageheight{8.5truein}
     \hsize\pagewidth
     \vsize\pageheight
     \stylenumber=3}
\def\refsize{\magnification=\magstep1
     \overfullrule0pt
     \def\pagewidth{390pt}
     \def\pageheight{9truein}
     \hsize\pagewidth
     \vsize\pageheight
     \stylenumber=4}
\def\rrversion{\magnification=\magstep1
     \overfullrule0pt
     \def\pagewidth{330pt}
     \def\pageheight{9truein}
     \hsize\pagewidth
     \vsize\pageheight
     \stylenumber=5}
\def\bigsize{\magnification=\magstep2
     \overfullrule0pt
     \def\pagewidth{325pt}
     \def\pageheight{9truein}
     \hsize\pagewidth
     \vsize\pageheight
     \stylenumber=6}
\def\bbigsize{\magnification=\magstep3
     \overfullrule0pt
     \def\pagewidth{271pt}
     \def\pageheight{9truein}
     \hsize\pagewidth
     \vsize\pageheight
     \stylenumber=7}
\def\USsmallprint{\magnification=\magstep0
     \overfullrule0pt
     \def\pagewidth{482pt}
     \def\pageheight{8.7truein}
     \hsize\pagewidth
     \vsize\pageheight
     \stylenumber=8}
\def\halfmagsize{\magnification=\magstephalf
     \def\pagewidth{427pt}
     \def\pageheight{9truein}
     \hsize\pagewidth
     \vsize\pageheight
     \stylenumber=9}
           % register  stylenumber  is used to control the output format

%addresses, letterheads
\def\deptaddress{\hskip 3truein Department of Mathematical Sciences\par
   \noindent e-mail: {\tt fremdh}\hskip 2.9truein\today\par
   \medskip}
\long\def\univaddress#1{\hskip 3truein Department of Mathematical Sciences\par
   \hskip 3truein University of Essex\par
   \hskip 3truein Colchester CO4 3SQ\par
   \hskip 3truein England\par
   %\hskip 3truein van Vleck Hall\par
   %\hskip 3truein 480 Lincoln Drive\par
   %\hskip 3truein Madison\par
   %\hskip 3truein WI 53706-1388, U.S.A.\par
   \hskip 3.2truein#1\par
   \noindent e-mail: {\tt david\@fremlin.org}
       %{\tt fremlin\@math.wisc.edu}
   \bigskip}
\long\def\homeaddress#1{\hskip 3truein 25 Ireton Road\par
   \hskip 3truein Colchester CO3 3AT\par
   \hskip 3truein England\par
   \hskip 3.2truein#1\par
   \noindent e-mail: {\tt david\@fremlin.org}
   \bigskip}
\long\def\letterhead#1#2#3{\univaddress{#1}
   {\parindent=0pt
    #2
   }
   \medskip

   \noindent Dear #3,
     }
\def\shortaddress{\noindent Departmentof Mathematical Sciences, University of Essex,
   Colchester CO4 3SQ, England\par
   \noindent e-mail: david\@fremlin.org}
\def\valediction#1#2{\bigskip
   \hskip 3truein #1,
   \vskip 1truein
   \hskip 3.3truein #2
   \bigskip}

%picture importation
   %It is clear that there is no general standard.   The syntax I
   %use when calling for a  .ps  file is
   %    \def\Caption{<...>}
   %    \picture{<filename>}{<dim>}
   %which is supposed to import  <filename>.ps  at height  <dim> ,
   %centered, and adding a one-line centered caption  <...> .
   %The basic macro listed in  \def\picture  is that which
   %used to work at Essex.   I now generally use the macro in
   % \def\picturewithpsfig , which requires a previous input
   % psfig.tex  or  psfig.sty ;
   %if you interpolate  \atUEssex  early in your file it will
   %give you the latter.
   %   I find that  psfig  is erratic in its centering, especially
   %under magnification, so I have written a routine
   % \centerpicturewithpsfig  (which may be called from
   % \centerpicture ) with the syntax
   %    \centerpicture{<filename>}{<dim>}{<dim>}
   %the first dimension being width and the second height.

   % A second package is called for by  \usegraphicx .   This requires
   % miniltx.tex  and  graphicx.sty ;  miniltx.tex  is currently (April 2016)
   % available from
   % http://mirror.ox.ac.uk/sites/ctan.org/macros/plain/graphics/miniltx.tex
   % The advantage of this is that it seems to allow manipulation of the
   % dvi>pdf stage using  dvipdfmx .

\def\picture#1#2{\bigskip
  \bigskip
  \immediate\write0{importing diagram #1.ps at height #2
         with dvitops}
  \vbox{\vskip #2
    \special{dvitops: import #1.ps \pagewidth #2}
    \Centerline{\Caption}
    }
   \medskip
   \def\Caption{}}
%an alternative version, without caption
\def\centerpicture#1#2#3{\bigskip
  \bigskip
  \immediate\write0{importing diagram #1.ps at width #2
         and height #3 with dvitops}
  \vbox{\vskip #3
    \special{dvitops: import #1.ps \pagewidth #3}
  }}
%to put pictures on same line
\def\sideshiftedpicture#1#2#3#4{\bigskip
   \bigskip
   \immediate\write0{importing diagram #1.ps at height #4
        and width #3 with dvitops}
   \vbox{\special{dvitops:  import #1.ps #3 #4}}
   \hskip #2 plus 0pt minus 0pt
   }
\def\startsideshiftedpicture{}
\newcount\scaleup
\newcount\scaledown
\scaleup=1
\scaledown=1
\def\scalesideshiftedpicture#1#2#3#4{
  \dimen4=#4
  \dimen3=#3
  \multiply\dimen4 by\scaleup
  \divide\dimen4 by \scaledown
  \multiply\dimen3 by\scaleup
  \divide\dimen3 by \scaledown
  \sideshiftedpicture{#1}{#2}{\dimen3}{\dimen4}}

\def\picturewithpsfig#1#2{\bigskip
   \immediate\write0{importing diagram #1.ps at height #2 with psfig}
   \vbox{\hsize\pagewidth
   \centerline{\psfig{figure=#1.ps,height=#2}}
   \medskip
   \centerline{\Caption}}
   \medskip
   \def\Caption{}}
\def\centerpicturewithpsfig#1#2#3{
  \immediate\write0{importing diagram #1.ps at width #2
         and height #3 with psfig}
  \dimen 3 = #2
  \dimen 4 = #3
  \dimen 2 = \pagewidth
  \advance\dimen2 by -\dimen3
  \divide\dimen2 by 2
  \discrversionB{\shrinkonestep}{}
  \discrversionH{\shrinkonestep}{}
  \discrversionF{\enlargeonestep}{}
  \discrversionG{\enlargeonestep\enlargeonestep}{}
  \dimen 6 = \dimen4
  \divide\dimen6 by 3
  \vskip\dimen6
  \ifdim\dimen2>0\hskip\dimen2\fi
     \vbox{\psfig{file=#1.ps,width=\dimen3,height=\dimen4}}
  \discrversionG{\vskip-\dimen6}{}
  }
\def\enlargeonestep{\dimen5 = 5truein
   \dimen6 = 7truein
   \ifdim\dimen3<\dimen5
      \ifdim\dimen4<\dimen6
         \divide\dimen3 by 5
         \multiply\dimen3 by 6
         \divide\dimen4 by 5
         \multiply\dimen4 by 6
      \fi\fi}
\def\shrinkonestep{\divide\dimen3 by 6
         \multiply\dimen3 by 5
         \divide\dimen4 by 6
         \multiply\dimen4 by 5}
\def\sideshiftedpicturewithpsfig#1#2#3#4{
   \immediate\write0{importing diagram #1.ps at height #4
      and width #3 with psfig}
   \dimen 4 = #4
   \dimen 3 = #3
   \discrversionB{\shrinkonestep}{}
   \discrversionH{\shrinkonestep}{}
   \discrversionF{\enlargeonestep}{}
   \discrversionG{\enlargeonestep\enlargeonestep}{}
%   \vbox{\psfig{figure=#1.ps,height=\dimen4,width=\dimen3}}
   \psfig{figure=#1.ps,height=\dimen4,width=\dimen3}
   \hskip #2 plus 0pt minus 0pt}

\def\picturewithgraphicx#1#2{\bigskip
   \immediate\write0{importing diagram #1.ps at height #2 with graphicx}
   \vbox{\hsize\pagewidth
   \centerline{\includegraphics[height=#2]{#1.ps}}
   \medskip
   \centerline{\Caption}}
   \medskip
   \def\Caption{}}
\def\centerpicturewithgraphicx#1#2#3{
  \immediate\write0{importing diagram #1.ps at width #2
         and height #3 with graphicx}
  \dimen 3 = #2
  \dimen 4 = #3
  \dimen 2 = \pagewidth
  \advance\dimen2 by -\dimen3
  \divide\dimen2 by 2
  \discrversionB{\shrinkonestep}{}
  \discrversionH{\shrinkonestep}{}
  \discrversionF{\enlargeonestep}{}
  \discrversionG{\enlargeonestep\enlargeonestep}{}
  \dimen 6 = \dimen4
  \divide\dimen6 by 3
  \vskip\dimen6
  \ifdim\dimen2>0\hskip\dimen2\fi
     \vbox{\includegraphics[width=\dimen3,height=\dimen4]{#1.ps}}
  \discrversionG{\vskip-\dimen6}{}
  }
\def\enlargeonestep{\dimen5 = 5truein
   \dimen6 = 7truein
   \ifdim\dimen3<\dimen5
      \ifdim\dimen4<\dimen6
         \divide\dimen3 by 5
         \multiply\dimen3 by 6
         \divide\dimen4 by 5
         \multiply\dimen4 by 6
      \fi\fi}
\def\shrinkonestep{\divide\dimen3 by 6
         \multiply\dimen3 by 5
         \divide\dimen4 by 6
         \multiply\dimen4 by 5}
\def\sideshiftedpicturewithgraphicx#1#2#3#4{
   \immediate\write0{importing diagram #1.ps at height #4
      and width #3 with graphicx}
   \dimen 4 = #4
   \dimen 3 = #3
   \discrversionB{\shrinkonestep}{}
   \discrversionH{\shrinkonestep}{}
   \discrversionF{\enlargeonestep}{}
   \discrversionG{\enlargeonestep\enlargeonestep}{}
%   \vbox{\includegraphics[height=\dimen4,width=\dimen3]{#1.ps}}
   \includegraphics[height=\dimen4,width=\dimen3]{#1.ps}
   \hskip #2 plus 0pt minus 0pt}

%alternative picture importation code;  I've been told about this
  %but have not used it successfully myself
%\def\Caption{}
%\input epsf
%\epsfverbosetrue
%\def\picture#1#2{\bigskip
%    \bigskip
%    \immediate\write0{importing diagram #1.ps at height #2 with epsf}
%    \epsfysize=#2
%    \centerline{\epsffile{#1.ps}}
%    \medskip
%    \Centerline{\Caption}
%    \medskip
%    }

%in addition it may be necessary to amend the imported  .ps  file
%by replacing the   %%BoundingBox   header by  %%BoundingBox:

\def\atUEssex{
   \input psfig.sty
   \def\picture##1##2{\picturewithpsfig{##1}{##2}}
   \def\sideshiftedpicture##1##2##3##4{
     \sideshiftedpicturewithpsfig{##1}{##2}{##3}{##4}}
   \def\centerpicture##1##2##3{
       \centerpicturewithpsfig{##1}{##2}{##3}}
   \def\startsideshiftedpicture{
       \sideshiftedpicture{empty}{1pt}{1pt}{1pt}}
   }
\def\usegraphicx{
   \input miniltx
   \def\Gin@driver{dvips.def}
   \input graphicx.sty
   \def\picture##1##2{\picturewithgraphicx{##1}{##2}}
   \def\sideshiftedpicture##1##2##3##4{
     \sideshiftedpicturewithgraphicx{##1}{##2}{##3}{##4}}
   \def\centerpicture##1##2##3{
       \centerpicturewithgraphicx{##1}{##2}{##3}}
   \def\startsideshiftedpicture{
       \sideshiftedpicture{empty}{1pt}{1pt}{1pt}}
}


%special fonts
\def\biggerscriptfonts{\scriptfont0=\eightrm
     \scriptfont1=\eighti
     \scriptfont2=\eightsy
     \scriptfont3=\eightex
     \scriptfont\bffam=\eightbf
     \scriptfont\msbfam=\eightmsb
     \scriptfont\eufmfam=\eighteufm
     }
\def\Loadcyrillic{\ifamstexsupported
      \input cyracc.def
      \font\tencyr=wncyr10
      \def\cyr{\tencyr\cyracc} \def\itcyr{\tenitcyr\cyracc}
    \else\def\cyr{\smc}\def\cdprime{u\i}\def\Cdprime{UI}\fi}
%http://wkweb3.cableinet.co.uk/linux/LDP/HOWTO/Cyrillic-HOWTO-7.html
%cyrillic.tex

\def\Loadfourteens{\iffourteensavailable
       \font\fourteenrm=cmr14
       \font\fourteenbf=cmbx14
       \font\fourteenit=cmsl14
   \else\font\fourteenrm=cmr10 scaled \magstep2
      \font\fourteenbf=cmbx10 scaled \magstep2
      \font\fourteenit=cmsl10 scaled \magstep2
   \fi}
\def\smallerfonts{\eightpoint
   \textfont\msbfam=\eightmsb
   \textfont\eufmfam=\eighteufm
   \textfont\bffam=\eightbf
   \textfont\eurmfam=\eighteurm
   \textfont\eusmfam=\eighteusm
   }
\font\twelveex=cmex10 scaled \magstep1


%Copyright declarations
\newcount\oldcopyrightdate
\newcount\newcopyrightdate
\oldcopyrightdate=0
\def\copyrightdate#1{\newcopyrightdate=#1
       \ifnum\newcopyrightdate=\oldcopyrightdate\else
       \oldcopyrightdate=#1
       \oldfootnote{}{\eightrm \copyright\ #1 D.~H.~Fremlin}
       \immediate\write0{Copyright #1}
       \fi}

%hyphenation
\hyphenation{equi-ver-id-ical}

%miscellaneous definitions
\def\Aut{\mathop{\text{Aut}}}
\def\backstep#1{\hskip -0.#1em plus 0 pt minus 0 pt}
\def\Bang{\vthsp\hbox{\bf X\hskip0.07em\llap X\hskip0.07em\llap X\rm}}
\def\BanG{\Bang{\hskip 1.111pt plus 3.333pt minus 0pt}}
%\def\BbbN{\mathchoice{\hbox{$\Bbb N$\hskip0.02em}}
%  {\hbox{$\Bbb N$\hskip0.02em}}
%  {\hbox{$\scriptstyle\Bbb N$\hskip0.04em}}
%  {\hbox{$\scriptscriptstyle\Bbb N$\hskip0.04em}}}
%\def\BbbQ{\mathchoice{\hbox{$\Bbb Q$\hskip0.02em}}
%  {\hbox{$\Bbb Q$\hskip0.02em}}
%  {\hbox{$\scriptstyle\Bbb Q$\hskip0.04em}}
%  {\hbox{$\scriptscriptstyle\Bbb Q$\hskip0.04em}}}
%\def\BbbR{\mathchoice{\hbox{$\Bbb R$\hskip0.02em}}
%  {\hbox{$\Bbb R$\hskip0.02em}}
%  {\hbox{$\scriptstyle\Bbb R$\hskip0.04em}}
%  {\hbox{$\scriptscriptstyle\Bbb R$\hskip0.04em}}}
%  %do not use with $\Bbb R^X$
%\def\BbbZ{\mathchoice{\hbox{$\Bbb Z$\hskip0.02em}}
%  {\hbox{$\Bbb Z$\hskip0.02em}}
%  {\hbox{$\scriptstyle\Bbb Z$\hskip0.04em}}
%  {\hbox{$\scriptscriptstyle\Bbb Z$\hskip0.04em}}}
\def\BbbN{\openBbb{N}}
\def\BbbQ{\openBbb{Q}}
\def\BbbR{\openBbb{R}}
  %do not use with $\Bbb R^X$  do not use in $_{\Bbb R^r}$
\def\BbbZ{\openBbb{Z}}
\def\Bbbone{\hbox{1\hskip0.13em\llap{1}}}
\def\biggerint{\textfont3=\twelveex\intop\nolimits}
  % you can get this size into a displayed formula with
  %  $$\hbox{$\biggerint_{\Cal S}\pmb{u}\,d\pmb{v}$}$$
  % but it may then seem cramped
  % or try \hbox{$\biggerint_{\Cal S}\,$}
\def\biggeriint{\textfont3=\twelveex\iint}
  % remarkable fact:  this seems to infect all \int within a given
  % $..$ ; but  $\biggerint$$\int$ seems to leave the $\int$ normal
  % really  \textfont3=\twelveex  is more reliable, since  \biggerint
  % as written gets turned off rather easily (e.g. by  \comment,
  % or by an extra  {...})
\def\bover#1#2{{{#1}\over{#2}}}
\def\Bover#1#2{\backstep4\hbox{
     \biggerscriptfonts${\shortstrut#1}\over{\shortstrut#2}$}}
  % when there are superscripts in the denominator, as in \Bover1{2^m}
  % or  \Bover1{\sqrt{2\pi}^r} , these can sometimes come out with better
  % horizontal alignment if you use  \Bover1{{\sshortstrut 2}^m}  or
  % \Bover1{{\sqrt{2\pi}}^r}
\def\bsover#1#2{\textstyle\bover{#1}{#2}}
\def\bsp{/\penalty-100}
  % \penalty seems to behave oddly in a \tt string in \fullsize mode
\def\Caption{}
\def\Center#1{

   \Centerline{#1}

   \noindent}
\def\Centerline#1{\vskip 5pt
     {\def\int{\text{}$$\textfont3=\twelveex\intop\nolimits}
      \def\dashint{\text{}$$\textfont3=\twelveex
         \mskip2mu\vrule height2.65pt width4.5pt depth-2.35pt\mskip-11mu
	 \intop\nolimits\mskip-2mu}
      \def\idashint{\text{}$$\int\mskip-7mu\dashint}	 
      \def\dashiint{\text{}$$\dashint\mskip-6mu\int}	 
      \def\dashidashint{\text{}$$\dashint\mskip-6mu\dashint}	 
      \def\underlineint{
           \text{}$$\textfont3=\twelveex\underline{\intop\nolimits}}
      \def\overlineint{
           \text{}$$\textfont3=\twelveex\overline{\intop\nolimits}}
      \def\iint{\text{}$$\textfont3=\twelveex
           \intop\nolimits\hskip-0.5em\intop\nolimits}
      \def\sum{\text{}$$\sumop}
      \def\prod{\text{}$$\prodop}
      \def\bigcap{\text{}$$\bigcapop}
      \def\bigcup{\text{}$$\bigcupop}
   \centerline{#1}}\vskip 5pt}
   \mathchardef\bigcapop="1354
   \mathchardef\bigcupop="1353
   \mathchardef\prodop="1351
   \mathchardef\sumop="1350
   %in  \Centerline{$\overline{}$, $\cmmnt{}$} must use \sumop etc
\def\cf{\mathop{\text{cf}}}
\def\circumflex#1{{\accent"5E #1}}
\def\closeplus{\hbox{+}}
\long\def\cmmnt#1{\ifresultsonly\else{#1}\fi}
\def\coint#1{\left[#1\right[}
\def\curl{\mathop{\text{curl}}}
\def\dashint{\mathchoice{-\mskip-20mu\intop\nolimits}
   {{\vrule height2.75pt width4.5pt depth-2.45pt\mskip-10mu\intop\nolimits}}
      {-\mskip-16.5mu\intop\nolimits} %not yet used
      {-\mskip-16.5mu\intop\nolimits}} %not yet used
\def\idashint{\mathchoice{\int\mskip-6mu\dashint}
      {\int\mskip-5mu\dashint}
      {\int\mskip-5mu\dashint} %not yet used
      {\int\mskip-5mu\dashint}} %not yet used
\def\dashiint{\mathchoice{\dashint\mskip-10mu\int}
      {\dashint\mskip-6mu\int}
      {\dashint\mskip-6mu\int} %not yet used
      {\dashint\mskip-6mu\int}} %not yet used
\def\dashidashint{\mathchoice{\dashint\mskip-10mu\dashint}
      {\dashint\mskip-6mu\dashint}
      {\dashint\mskip-6mu\dashint} %not yet used
      {\dashint\mskip-6mu\dashint}} %not yet used
\def\dashidashiint{\mathchoice{\dashint\mskip-8mu\dashint\mskip-10mu\int}
      {\dashint\mskip-4mu\dashint\mskip-8mu\int} 
      {\dashint\mskip-6mu\int} %not yet used
      {\dashint\mskip-6mu\int}} %not yet used
\def\dashiidashint{\mathchoice{\dashint\mskip-10mu\int\mskip-8mu\dashint}
      {\dashint\mskip-4mu\dashint\mskip-8mu\int} 
      {\dashint\mskip-6mu\int} %not yet used
      {\dashint\mskip-6mu\int}} %not yet used
\def\idashidashint{\mathchoice{\int\mskip-8mu\dashint\mskip-8mu\dashint}
      {\dashint\mskip-6mu\int} %not yet used
      {\dashint\mskip-6mu\int} %not yet used
      {\dashint\mskip-6mu\int}} %not yet used
\def\diam{\mathop{\text{diam}}}
\def\discrpage{\ifnum\stylenumber=1{\frnewpage}  %redefined in mt.tex
           \else\ifnum\stylenumber=3{\frnewpage}
           \else\ifnum\stylenumber=5{\frnewpage}
           \else\bigskip\fi\fi\fi}
\long\def\discrversionA#1#2{\ifnum\stylenumber=1#1\else#2\fi}
\long\def\discrversionB#1#2{\ifnum\stylenumber=2#1\else#2\fi}
\long\def\discrversionC#1#2{\ifnum\stylenumber=3#1\else#2\fi}
\long\def\discrversionD#1#2{\ifnum\stylenumber=4#1\else#2\fi}
\long\def\discrversionE#1#2{\ifnum\stylenumber=5#1\else#2\fi}
\long\def\discrversionF#1#2{\ifnum\stylenumber=6#1\else#2\fi}
\long\def\discrversionG#1#2{\ifnum\stylenumber=7#1\else#2\fi}
\long\def\discrversionH#1#2{\ifnum\stylenumber=8#1\else#2\fi}
           % facility for special interpolations in particular output
           % versions
\def\displaycause#1{\noalign{\noindent (#1)}}
\def\diverg{\mathop{\text{div}}}
\def\dom{\mathop{\text{dom}}}
\def\eae{=_{\text{a.e.}}}

\def\esssup{\mathop{\text{ess sup}}}
\mathchardef\Exists="3239

\def\family#1#2#3{\langle#3\rangle_{#1\in#2}}
\def\familyi#1#2{\family{i}{#1}{#2}}
\def\familyiI#1{\familyi{I}{#1}}

\def\frfilename#1{\ifnum\stylenumber=1{\noindent filename #1}
               \else\ifnum\stylenumber=6{\noindent{\fiverm filename #1}}
               \else\ifnum\stylenumber=7{\noindent{\fiverm filename #1}}
               \else\fi\fi\fi}
\def\Footnote#1#2{\footnote"#1"{#2}}
\mathchardef\Forall="3238
\def\frakc{\frak c\phantom{.}}
\def\frnewpage{\vskip \pageheight plus 0pt minus\pageheight
           \eject
           \gdef\topparagraph{}
           \gdef\bottomparagraph{}
           }
\def\frsmallcirc{{\hskip 1pt plus 0pt minus 0pt}
     {\raise 1pt\hbox{$\smallcirc$}}{\hskip 1pt plus 0pt minus 0pt}}

\def\geae{\ge_{\text{a.e.}}}
\def\grad{\mathop{\text{grad}}}
\def\grheada{{\bf ($\pmb{\alpha}$)}}
\def\grheadb{{\bf ($\pmb{\beta}$)}}
\def\grheadc{{\bf ($\pmb{\gamma}$)}}
\def\grheadd{{\bf ($\pmb{\delta}$)}}
\def\grheade{{\bf ($\pmb{\epsilon}$)}}
\def\grheadz{{\bf ($\pmb{\zeta}$)}}
\def\grv#1{{\accent"12 #1}}

\def\hfillall{\hskip\pagewidth plus0pt minus\pagewidth}

\def\Imag{\mathop{\Cal I\text{m}}}
\long\def\inset#1{{\narrower\narrower{\vskip 1pt plus 1pt minus 0pt}
     #1
     {\vskip 1pt plus 1pt minus 0pt}}}
\def\Int#1#2{\displaystyle{\int_{#1}^{#2}}}
\def\interior{\mathop{\text{int}}}

\def\leae{\le_{\text{a.e.}}}
\long\def\leaveitout#1{}
\def\LLcorner{\mathchoice{\,\hbox{\vrule height7pt width0.5pt depth0pt
       \vrule height0.5pt width5pt depth0pt}\,}
   {\,\hbox{\vrule height7pt width0.5pt depth0pt
       \vrule height0.5pt width5pt depth0pt}\,}
   {\,\hbox{\vrule height5pt width0.3pt depth0pt
       \vrule height0.3pt width4pt depth0pt}\,}
   {\,\hbox{\vrule height4pt width0.25pt depth0pt
       \vrule height0.25pt width3pt depth0pt}\,}
  }
\def\Ln{\mathop{\text{Ln}}}
\def\Loadfont#1{\font#1}
\def\loibr{\hbox{\hskip0.1em]\hskip0.05em}}
\def\Matrix#1{\pmatrix#1\endpmatrix}
\def\Mu{\text{M}}
\def\noamslogo{\def\amslogo{}}
\def\normalsubgroup{\vartriangleleft}
\def\Nu{\text{N}}
\def\ocint#1{\left]#1\right]}
\def\ofamily#1#2#3{\langle#3\rangle_{#1<#2}}
\let\oldcases=\cases
\let\oldfootnote=\footnote
\def\On{\mathop{\text{On}}}
\def\ooint#1{\left]#1\right[}
\def\openBbb#1{\Bbb#1\mskip1mu}
\def\pd#1#2{\bover{\partial#1}{\partial#2}}
\def\Pd#1#2{\Bover{\partial#1}{\partial#2}}
\def\pdd#1#2{{{\partial^2#1}\over{\partial#2^2}}}
\def\Pdd#1#2{\Bover{\partial^2#1}{\partial#2^2}}
\def\pde#1#2#3{{{\partial^2#1}\over{\partial#2\partial#3}}}
\def\Pde#1#2#3{\Bover{\partial^2#1}{\partial#2\partial#3}}
\def\Prf{\vthsp\hbox{\bf P\hskip0.07em\llap P\hskip0.07em\llap P\rm}}
\def\pushbottom#1{\hbox{\vrule height0pt depth#1 width0pt}}
\def\Qed{\vthsp\hbox{\bf Q\hskip0.07em\llap Q\hskip0.07em\llap Q\rm}}
\def\QeD{\Qed{\hskip 1.111pt plus 3.333pt minus 0pt}}
\def\Quer{\vthsp\hbox{\bf ?\hskip0.05em\llap ?\hskip0.05em\llap
   ?\thinspace\rm}}
\def\query{\discrversionA{\immediate\write0{query}
    \global\advance\footnotenumber by 1
    \oldfootnote{$^{\the\footnotenumber}$}{query}}{}}
\def\Real{\mathop{\Cal R\text{e}}}
\def\restr{\mathchoice
  {\hbox{$\restriction$}\mskip 1mu}
  {\hbox{$\restriction$}\mskip 1mu}
  {\hbox{$\scriptstyle\restriction$}\mskip 1mu}
  {\hbox{$\scriptscriptstyle\restriction$}\mskip 1mu}}
\def\restrp{\mathchoice
  {\hbox{$\restriction$}\mskip 2mu}
  {\hbox{$\restriction$}\mskip 2mu}
  {\mskip-1mu\hbox{$\scriptstyle\restriction$}\mskip 2mu}
  {\mskip-1mu\hbox{$\scriptscriptstyle\restriction$}\mskip 1mu}}
    %use in \restrp V \restrp W \restrp Y
    % \restrp\Cal P \restrp\Bbb R \restrp\frak
    % \restrp\Omega \restrp\Tau \restrp\Upsilon
    % \restrp[ \restrp\widehat{}
\def\Rho{\text{P}}
\def\roibr{\hbox{\hskip0.05em[\hskip0.1em}}
\def\RoverC{\hbox{\biggerscriptfonts${{\Bbb R}\atop{\Bbb C}}$}}
\def\sequence#1#2{\family{#1}{\Bbb N}{#2}}
\def\sequencen#1{\sequence{n}{#1}}
\def\setover#1#2{\overset{#1}\to{#2}}
\def\shortstrut{\hbox{\vrule height7.5pt depth2.5pt width0pt}}
\def\sshortstrut{\hbox{\vrule height5.8pt depth2.5pt width0pt}}
\def\smallcirc{{\scriptstyle\circ}}
\def\ssbullet{{\scriptscriptstyle\bullet}}
\def\symmdiff{\triangle}
\def\thicken#1{#1\hskip0.03em\llap #1\hskip0.03em\llap
   #1\hskip0.03em\llap #1}
\def\Tau{{\text{T}}}
\def\tbf#1{\text{\bf #1}}
\def\terror#1#2#3{#1:  `#2' for `#3'}
\def\today{\number\day.\number\month.\number\year}
\def\UEMDRR{University of Essex Mathematics Department Research Report}
\def\UEMDWP{University of Essex Mathematics Department Working Paper}
\def\versiondate#1{
    \immediate\write0{Version of #1}
    \def\vsdate{#1}
               \ifnum\stylenumber=1\hfill Version of #1
                              \medskip
               \else\ifnum\stylenumber=2\hfill Version of #1
                              \medskip
               \else\ifnum\stylenumber=6\hfill{\fiverm Version of #1}
                              \medskip
               \else\ifnum\stylenumber=7\hfill{\fiverm Version of #1}
                              \medskip
               \else\ifnum\stylenumber=8\hfill Version of #1
                              \medskip
               \else\fi\fi\fi\fi\fi}
\def\versionhead{\ifnum\stylenumber=3 \hfill PRINTER'S VERSION

                        \bigskip
                   \else\ifnum\stylenumber=4 \hfill REFEREE'S VERSION

                        \bigskip
                   \else \fi\fi}
\def\versionref{\ifnum\stylenumber=1\workingref
           \else\ifnum\stylenumber=2\workingref
           \else\ifnum\stylenumber=8\workingref
           \else\printerref\fi\fi\fi}
\def\vthsp{\hskip0.001em}

%Missing AmSTeX can be coped with by using the following lines:

\ifamstexsupported\else
   \immediate\write0{AmSTeX-unsupported version of  fremtex.tex }
   \fourteensavailablefalse
   \def\@{@}
   \def\\{\cr}
   \def\AmSTeX{AmS\TeX}
   \def\Bbb#1{\underline{\hbox{\bf #1}}}
   \def\biggerscriptfonts{}
   \def\bumpeq{=_{\text{approx}}}
   \def\Cal#1{{\cal#1}}
   \def\curlyeqprec{\preceq}
   \def\eightit{\it}
   \def\eightrm{\rm}
   \def\eightsmc{\smc}
   \def\eurm#1{\underline{\cal#1}}
   \def\Footnote#1#2{\footnote#1#2}
   \def\frac#1#2{{#1}\over{#2}}
   \def\frak#1{{\hbox{\bf #1\hskip0.1em\llap #1}}}
   \def\frnewpage{\vfill\eject}
   \def\iint{\int\!\!\!\int}
   \def\iiint{\int\!\!\!\int\!\!\!\int}
   \def\iiiint{\int\!\!\!\int\!\!\!\int\!\!\!\int}
   \def\llcorner{\lfloor}
   \def\loadeurm{}
   \def\loadeusm{}
%   \def\loadeusmx#1{\def\eusm#1{\underline{\cal#1}}}
   \def\ltimes{|\hskip-0.5em\times}
   \def\eusm#1{\underline{\cal#1}}
   \def\ltimes{|\times}
   \def\Loadfont#1{}
   \def\Matrix#1{\pmatrix{#1}}
   \def\noamslogo{}
   \def\nrightarrow{\not\rightarrow}
   \def\preccurlyeq{\preceq}
   \def\query{\discrversionA{\immediate\write0{query}
                             \footnote{$^*$}{query}}{}}
   \def\restriction{|}
   \def\rtimes{\times|}
   \def\setover#1#2{#2^{#1}}
   \def\shortstrut{\strut}
   \def\smallerfonts{\eightrm}
   \def\smallfrown{\wedge}
   \def\smc{}
   \def\spcheck{\check{\phantom{x}}}
   \def\sphat{\hat{\phantom{x}}}
   \def\square{\mathchoice\sqr64\sqr64\sqr43\sqr33}
   \def\sqr#1#2{{\vcenter{\vbox{\hrule height.#2pt
          \hbox{\vrule width.#2pt height#1pt \kern#1pt
                  \vrule width.#2pt}
          \hrule height.#2pt}}}}
   \def\pmb#1{\underline{#1}}
   \def\text#1{\hbox{#1}}
   \def\therefore{\ssbullet\raise.9ex\hbox{$\ssbullet$}\ssbullet}
   \def\twelvebf{\bf}
   \def\vartriangleleft{\triangleleft}
   \def\Vdash{|\vdash}
   \def\vDash{|=}
   \fi

%End of lines for missing AmSTeX

\ifamstexsupported\input amstex
%              \documentstyle{amsppt}
               \input amsppt.sty
               \PSAMSFonts
	       \tenpoint
               %\font\eightmsb=msym8
               \font\eightmsb=msbm10
               \font\eighteufm=eufm10 at 8 pt
               \font\eighteurm=eurm8
               \font\eighteusm=eusm8
               \font\varseveneufm=eufm10 at 7 pt
               \scriptfont\eufmfam=\varseveneufm
   \def\.#1{{\accent"5F #1}}
   \def\=#1{{\accent"16 #1}}
   \gdef\nopagenumbers{\NoPageNumbers}
   \else\fi

\newcount\footmarkcount
\ifDHFamstexloaded
    \loadmsBm
    \def\eightBbb#1{{\fam\msBfam#1}}
    \loadmsFm
    \def\eightfrak#1{{\fam\msFfam#1}}
  \else
    \def\eightBbb#1{{\scriptstyle\Bbb#1}}
    \def\eightfrak#1{{\scriptstyle\frak#1}}
  \fi
\ifathome\athome\fi

%interword glue (\tenrm standard):
%  (space) 3.333 + 1.667 - 1.111
%  (,)     3.333 + 2.083 - 0.889
%  (.)     4.444 + 5.    - 0.370

%very odd fact:  \discrversionA{2.}{}  doesn't work,
%   \discrversionA{ 2.}{} does

%The false line feed introduced by my Fortran programs, leading to
%a surplus  ae  at the top of the next page, seems to be associated
%with  Fk(\032)3081  or something similar in the PostScript file.


\immediate\write0{Extract from MEASURE THEORY,
   by D.H.Fremlin, University of Essex, Colchester.
   This material is copyright.   It is issued under the terms of the
   Design Science License as published in
   http://dsl.org/copyleft/dsl.txt.   This is a development version and the
   source files are not permanently archived.
   For further information contact david\@fremlin.org.}

%hyphenations
\hyphenation{Archi-me-dean}

%new style
\def\smallprintersize{\magnification=\magstep0
     \overfullrule0pt
     \def\pagewidth{468pt}
     \def\pageheight{9truein}
     \hsize\pagewidth
     \vsize\pageheight
     \nologo
     \def\copyrightdate##1{}
     \def\folio{\ifnum\pageno<0{}\else\number\pageno\fi}
     \def\mtfootline{}
     \stylenumber=10
     \pageno=-1000}

%With-proofs, results-only versions
\newif\ifwithproofs
\withproofstrue
\newif\ifresultsonly
\resultsonlyfalse
\def\noproofs{\withproofsfalse}
\def\resultsonly{\resultsonlytrue\withproofsfalse}
\long\def\proof#1{\ifwithproofs{\medskip

     \noindent{\bf proof} #1}\else\fi}
\long\def\prooflet#1{\ifwithproofs{#1}\else\fi}
%\long\def\Prooflet#1{\prooflet{\Prf\ #1 \Qed\ }}
\long\def\endnotes#1{\ifresultsonly\else{#1}\fi}
\long\def\exercises#1{\ifresultsonly\else{#1}\fi}
\long\def\references#1{\ifresultsonly\else{#1}\fi}
\long\def\dvro#1#2{\ifresultsonly{#1}\else{#2}\fi}

%Headline assignments

\def\topparagraph{}
\def\bottomparagraph{}
\def\newparagraph{}
\newif\ifpageturned

\newdimen\rhoffset
\newdimen\lhoffset
\lhoffset=0cm
\rhoffset=0cm

\newcount\volumeno
\volumeno=5
\newcount\luluvolumeno
\luluvolumeno=0

\let\plainpagebody=\pagebody
\def\pagebody{\ifodd\pageno\global\hoffset=\rhoffset
         \else\global\hoffset=\lhoffset\fi
     \plainpagebody
     \turnpage}
\def\turnpage{
     \iffirstpage\else{\baselineskip=24pt\line{\mtfootline}}\fi
%     \immediate\write0{turning page \folio,
%           top paragraph \topparagraph,   bottom \bottomparagraph,
%           new \newparagraph, pagetotal \the\pagetotal}
     \global\pageturnedtrue
     \global\firstpagefalse
     \global\let\topparagraph=\bottomparagraph
     \global\let\bottomparagraph=\newparagraph
     }

\newdimen\pagetotalplus
\newdimen\pagetotalminus

%wheader parameters:  #1 new paragraph name
 % #2#3#4  \vskip parameters if page break not forced (no. of points)
 % #5  spare height demanded if page break not to be forced
\long\def\wheader#1#2#3#4#5{\gdef\newparagraph{#1}
  \pagetotalplus=\the\pagetotal
  \advance\pagetotalplus by #5
  \pagetotalminus=\the\pagetotal
  \advance\pagetotalminus by -\pageshrink
  %\showthe\pagetotal \showthe\pageshrink \showthe\pagegoal
  \global\pageturnedfalse
  \ifdim\pagetotalminus<\pagegoal\ifdim\pagetotalplus>\pagegoal
%     \immediate\write0{forcing page break}
     \eject
     \else\vskip #2pt plus #3pt minus #4pt\fi
     \else\vskip #2pt plus #3pt minus #4pt\fi
  \ifpageturned\global\let\topparagraph=\newparagraph
%     \immediate\write0{paragraph change on turning page}
     \fi
%  \immediate\write0{top paragraph \topparagraph, bottom
%       \bottomparagraph, pagetotal \the\pagetotal}
  \ifdim\pagetotal>\pagegoal\else\gdef\bottomparagraph{#1}\fi
%   \immediate\write0{top paragraph \topparagraph, bottom
%       \bottomparagraph, pagetotal \the\pagetotal}
  }
\def\header#1{
     \wheader{#1}{10}{4}{4}{24pt}}
\def\leader#1#2{\immediate\write0{#1 #2}
     \wheader{#1}{14}{7}{5}{24pt}
     {\bf #1 #2}}
\def\vleader#1#2#3{\immediate\write0{#2 #3}
     \wheader{#2}{14}{7}{5}{#1}
     {\bf #2 #3}}
\def\Notesheader#1{\immediate\write0{#1 Notes and comments}
     \wheader{#1 Notes}{16}{8}{5}{48pt}
     \noindent{\bf #1 Notes and comments}}
\def\vNotesheader#1#2{\immediate\write0{#2 Notes and comments}
     \wheader{#2 Notes}{16}{8}{5}{#1}
     \noindent{\bf #2 Notes and comments}}
\def\spheader#1#2#3#4#5{\header{#1#2#3#4#5}{\bf (#5)}}
\def\spstheader#1#2#3#4#5{\header{*#1#2#3#4#5}{\bf (#5)}}
\def\vspheader#1#2#3#4#5#6{
     \wheader{#2#3#4#5#6}{10}{4}{4}{#1}{\bf (#6)}}
\def\sqheader#1#2#3#4#5{\header{#1#2#3#4#5}{$\pmb{>}${\bf (#5)}}}
\def\errheader#1#2#3{
     \wheader{#3}{6}{4}{2}{24pt}
     {\bf p #1 l #2}}
\long\def\woddheader#1#2#3#4#5{
   \ifodd\pageno\wheader{#1}{#2}{#3}{#4}{#5}
   \else\vskip #2pt plus #3pt minus #4pt\fi}

\def\indexmedskip{
  \pagetotalplus=\the\pagetotal
  \advance\pagetotalplus by 18 pt
  \pagetotalminus=\the\pagetotal
  \advance\pagetotalminus by -\pageshrink
  \ifdim\pagetotalminus<\pagegoal\ifdim\pagetotalplus>\pagegoal
      \else\medskip\fi\else\medskip\fi}

\def\indexheader#1{\immediate\write0{#1}\vindexheader{#1}{24}}
\def\vindexheader#1#2{
  \gdef\newparagraph{#1}
  \pagetotalplus=\the\pagetotal
  \advance\pagetotalplus by #2 pt
  \pagetotalminus=\the\pagetotal
  \advance\pagetotalminus by -\pageshrink
  %\showthe\pagetotal \showthe\pageshrink \showthe\pagegoal
  \global\pageturnedfalse
%  \ifdim\pagetotal>\pagegoal
%      \immediate\write0{warning: pagetotal greater than pagegoal}
%      \immediate\write0{...   last paragraph reached is #1}
%      \immediate\write0{...   check for underfull vbox here}
%      \else\fi
  \ifdim\pagetotalminus<\pagegoal\ifdim\pagetotalplus>\pagegoal{\eject}
      \fi\fi
  \ifdim\pagetotal>\pagegoal\else\gdef\bottomparagraph{#1}\fi
  \ifpageturned{\global\let\topparagraph=\newparagraph
%     \immediate\write0{paragraph change on turning page}
      }\else\fi
%  \immediate\write0{top paragraph \topparagraph, bottom
%    \bottomparagraph}
  }

\def\leftheadline{\rlap{\folio}\hfill{\eightit\chaptername}\hfill
       \llap{\eightbf\topparagraph}}
\def\rightheadline{\rlap{\eightbf\bottomparagraph}\hfill
       {\eightit\headlinesectionname}\hfill\llap{\folio}}

%Footline assignments
\def\mtfootline{\ifodd\pageno\rightfootline\else\leftfootline\fi}
\def\rightfootline{\hfill\eightsmc D.H.Fremlin}
\def\leftfootline{\ifwithproofs{\eightsmc Measure Theory\hfill}
    \else{\eightsmc Measure Theory \eightit (abridged version)\hfill}
    \fi}

%Style assignments
\def\newchapter#1{\gdef\topparagraph{}
   \gdef\bottomparagraph{Chap.\ #1 {\it intro.}}
   \gdef\newparagraph{Chap.\ #1 {\it intro.}}
   \footmarkcount=0
   \gdef\sectionname{Introduction}
   \gdef\headlinesectionname{Introduction}
   \centerline{\bf Chapter #1}
   \medskip
   \centerline{\bf \chaptername}
   \medskip
   }
\def\newsection#1{\gdef\bottomparagraph{\S#1 {\it intro.}}
   \noindent{\bf #1 \sectionname}
   \smallskip
   \let\headlinesectionname=\sectionname}
\def\newvolume#1{\gdef\topparagraph{}
   \gdef\bottomparagraph{Vol.\ #1 {\it intro.}}
   \Loadfont{\twelvebf=cmbx12}
   \def\chaptername{Volume #1}
   \def\sectionname{\volumename}
   \def\headlinesectionname{\volumename}
   \centerline{\twelvebf Volume #1:  \volumename}
   \bigskip
   }

%font variations
\newif\ifvolwp
\volwpfalse
\def\tenrmstretch#1{\fontdimen3\tenrm=#1}
\font\elevenex=cmex10 at 11pt

%contents and introductions
\newif\iflargelogo
\def\vtmpb#1{}
\def\newvolume#1{}
\def\pagereference#1#2{\hfill\hbox{\ifresultsonly#2\qquad \hskip 1truein
                                                 \else#1\qquad \fi}}
\long\def\section#1#2#3#4#5#6{\ifresultsonly
    \qquad #1 #2\pagereference{#4}{#5}
  \else\vskip 2pt plus 1pt minus 0pt
    \vbox{\qquad #1 #2\pagereference{#4}{#5}\par
    \abstr{#6}
    }\vskip -1pt plus 1pt minus 0pt
  \fi}
\long\def\chapintrosection#1#2#3{\ifresultsonly
    \qquad Introduction\pagereference{#2}{#3}
  \else\vskip 1pt plus 1pt minus 0pt
    \qquad Introduction\pagereference{#2}{#3}
    \vskip 1pt plus 1pt minus 0pt
  \fi}
\def\abstr#1{\ifresultsonly{}
   \else{\advance\leftskip by\parindent
     \advance\rightskip by\parindent
     \advance\leftskip by\parindent
     \advance\rightskip by\parindent
     \advance\leftskip by\parindent
     \advance\rightskip by\parindent
     \advance\leftskip by\parindent
     \vskip1pt plus 0pt minus0pt
     \parindent=0pt
     \smallerfonts #1%
     \hbox{\vrule height0pt depth2pt width0pt}
     \vskip1pt plus 0pt minus0pt}\fi}
     
%language variations
\newif\ifUSEnglish
\USEnglishfalse
%\USEnglishtrue

%"Later editions only"
\newcount\latereditionpage
\newcount\latereditionmark
\latereditionpage=-1
\latereditionmark=0
\newcount\newpageno

%removing the \write0 instruction altogether disorganizes it!
\def\Latereditions{\newpageno=\pageno%
  \ifdim\pagetotal>\pagegoal\advance\newpageno by 1\fi%
     \divide\newpageno by 2%
  \immediate\write0{%
%     "Later editions only";
%      pagetotal \the\pagetotal, pagegoal \the\pagegoal,
%      pageno \number\pageno,\par newpageno \number\newpageno,
%      latereditionpage \number\latereditionpage%
  }\ifnum\latereditionpage=\newpageno%
     \ifDHFamspptloaded$^{\number\latereditionmark}$%
     \else\footnote{Later editions only.}\fi%
  \else\footnote{Later editions only.}%
  \global\latereditionpage=\newpageno%
  \global\latereditionmark=\footmarkcount\fi}%
%  \immediate\write0{newpageno \number\newpageno,
%      latereditionpage \number\latereditionpage}

   
%general definitions
\def\action{\mathchoice{\lower-.3ex\hbox{$\ssbullet$}}
                       {\lower-.3ex\hbox{$\ssbullet$}}
                       {\ssbullet}
                       {\ssbullet}}  %see also \varaction
\def\add{\mathop{\text{add}}\nolimits}
\long\def\allowmorestretch#1#2{
  \ifdim\pagewidth<#1pt\else
    \fontdimen3\tenrm=2pt\fontdimen4\tenrm=1.67pt\fi
  #2
    \fontdimen3\tenrm=1.67pt\fontdimen4\tenrm=1.11pt}
\def\am{atom\discretionary{-}{}{}lessly-measurable}
\def\AmuA{\Aut_{\bar\mu}\mskip-2mu\frak A}
\def\ASint{\mathchoice{\Bbb A\mskip-5mu}
                  {\Bbb A\mskip-3mu}
                  {\Bbb A\mskip-2mu}
                  {\Bbb A\mskip-2mu}\Sint}
\def\Asisfour{$(\frak A,\bar\mu,T,\family{t}{T}{\frak A_t})$}
\def\Asissix{$(\frak A,\bar\mu,T,
   \penalty-100\family{t}{T}{\frak A_t},
   \penalty-100\Cal T,\family{\tau}{\Cal T}{\frak A_{\tau}})$}
\def\Asisfourt{$(\frak A,\bar\mu,\coint{0,\infty},
   \langle\frak A_t\rangle_{t\ge 0})$}
\def\Asissixt{$(\frak A,\bar\mu,\coint{0,\infty},
   \penalty-100\langle\frak A_t\rangle_{t\ge 0},
   \penalty-100\Cal T,\family{\tau}{\Cal T}{\frak A_{\tau}})$}
\def\Atop#1#2{{\scriptstyle #1}\atop{\scriptstyle #2}}
\def\Aut{\mathop{\text{Aut}}}

\def\Bcap{\Booleanop{\cap}}
\def\Bcup{\Booleanop{\cup}}
\def\Booleanop#1{
   \ifamstexsupported{\mathchoice
    {\hbox{\hskip.2em\scriptfont2=\eightsy$\scriptstyle#1$\hskip.2em}}
    {\hbox{\hskip.2em\scriptfont2=\eightsy$\scriptstyle#1$\hskip.2em}}
 %   {{\mskip2mu\scriptstyle#1\mskip2mu}}  %until 15.10.17
    {{\scriptscriptstyle#1}}  %from 15.10.17
    {{\scriptscriptstyle#1}}}
   \else{\mathchoice{{\scriptstyle#1}}{{\scriptstyle#1}}
    {{\scriptstyle#1}}{{\scriptscriptstyle#1}}}
    \fi}
\def\Bsetminus{\Booleanop{\setminus}}
\def\Bsubseteq{\hskip.1em\Booleanop{\subseteq}\hskip.1em}
\def\Bsubset{\Booleanop{\subset}}
\def\Bsupseteq{\Booleanop{\supseteq}}
\def\Bsupset{\Booleanop{\supset}}
\def\Bsymmdiff{\Booleanop{\symmdiff}}
\def\Bcapshort{\Booleanopshort{\cap}}
\def\Bcupshort{\Booleanopshort{\cup}}
\def\Booleanopshort#1{
   \ifamstexsupported{\mathchoice
    {\hbox{\scriptfont2=\eightsy$\scriptstyle#1$}}
    {\hskip-1pt\hbox{\scriptfont2=\eightsy$\scriptstyle#1$}}
    {{\scriptstyle#1}}
    {{\scriptscriptstyle#1}}}
   \else{\mathchoice{{\scriptstyle#1}}{{\scriptstyle#1}}
    {{\scriptstyle#1}}{{\scriptscriptstyle#1}}}
    \fi}
\def\Bsetminusshort{\Booleanopshort{\setminus}}
\def\BSint{\mathchoice{\Bbb B\mskip-5mu}
                  {\Bbb B\mskip-3mu}
                  {\Bbb B\mskip-2mu}
                  {\Bbb B\mskip-2mu}\Sint}
%\def\Bsubseteqshort{\hskip.1em\Booleanopshort{\subseteq}\hskip.1em}
\def\Bsubseteqshort{\hskip.1em\Booleanopshort{\subseteq}}
\def\Bsubsetshort{\Booleanopshort{\subset}}
\def\Bsupseteqshort{\Booleanopshort{\supseteq}}
\def\Bsymmdiffshort{\Booleanopshort{\symmdiff}}

\def\closeBooleanop#1{
   \ifamstexsupported{\mathchoice
    {\hbox{\scriptfont2=\eightsy$\scriptstyle#1$}}
    {\hbox{\scriptfont2=\eightsy$\scriptstyle#1$}}
    {{\scriptstyle#1}}
    {{\scriptscriptstyle#1}}}
   \else{\mathchoice{{\scriptstyle#1}}{{\scriptstyle#1}}
    {{\scriptstyle#1}}{{\scriptscriptstyle#1}}}
    \fi}
\def\closeBcup{\closeBooleanop{\cup}}
\def\closeBcap{\closeBooleanop{\cap}}
\def\closeBsetminus{\closeBooleanop{\setminus}}
\def\closeBsymmdiff{\closeBooleanop{\symmdiff}}

\def\biggerHeint{\textfont3=\twelveex
   \raise 0.6pt\hbox to 0pt{\hskip 2.6pt{\sevenrm H}\hss}
   \intop\nolimits}
\def\biggerPfint{\textfont3=\twelveex
   \raise 0.6pt\hbox to 0pt{\hskip 1.3pt{\sevenrm Pf}\hss}
   \intop\nolimits\hskip 1.2pt}
\def\biggerSint{\textfont3=\twelveex
   \raise 0.6pt\hbox to 0pt{\hskip 3.6pt{\sevenrm S}\hss}
   \intop\nolimits}
\def\bu{\mathop{\text{bu}}}
\def\Bvalue#1{\mathchoice
   {\hbox{$[\![#1]\!]$}}
   {\hbox{$[\![#1]\!]$}}
   {\hbox{$\scriptstyle[\![#1]\!]$}}
   {\hbox{$\scriptscriptstyle[\![#1]\!]$}}}
\def\cadlag{c\`adl\`ag}
\def\CalBa{\Cal{B}\eurm{a}}
   %note: neither  \cal  nor  \eusm  has lower case
\def\CalFr{\Cal F_{\text{Fr}}}
\def\callal{c\`all\`al}
\def\CalNwd{\Cal{N}\hskip-1.2pt\eurm{w}\hskip-.8pt\eurm{d}}

\def\CalRbg{\Cal{R}\mskip-1mu\eurm{b}\mskip-1mu\eurm{g}}
\def\CalSmz{\Cal{S}\mskip-2mu\eurm{mz}}
\def\CalUn{\Cal{U}\eurm{n}}
\def\capacity{\mathop{\text{cap}}\nolimits}
\def\Caratheodory{Cara\-th\'eo\-dory}
\def\Cdlg{C_{\text{dlg}}}
\def\cff{\cf\mskip1mu}
\def\checkpmb#1{\check{\pmb #1}}
\def\ci{\mathop{\text{ci}}}
\def\cl{\mathop{\text{cl}}}
\def\Clll{\hbox{$C$
  \hskip-4.5pt\raise4pt\hbox{\char'23
  \hskip-1.6pt\vrule height6.5pt width.5pt depth0pt
  \hskip.7pt\vrule height6.5pt width.5pt depth0pt
  \hskip-2.5pt\lower5.4pt\hbox{\char'23}}}}
\def\clMsimp{\overline{M}_{\text{simp}}}
\def\closeuparrow{\mathchar"0222}
\def\closedownarrow{\mathchar"0223}
\def\clstar{\text{cl*}}
\def\cov{\mathop{\text{cov}}}
\def\covar#1#2{\mathchoice
  {[#1{\lower4pt\hbox{${\scriptstyle *}\atop{\vrule height 18pt depth -10pt}$}}#2]}
  {[#1{*\atop{\vrule height 8pt depth 0pt}}#2]}
  {[#1{*\atop{\vrule height 6pt depth 0pt}}#2]}
  {[#1{*\atop{\vrule height 6pt depth 0pt}}#2]}}
\def\covSh{\mathop{\text{cov}}\nolimits_{\text{Sh}}}
\def\cdownarrow{c\mskip2mu^{\downarrow}}
\def\cycle#1{(\overleftarrow{#1})}
\def\ddaggerloc{$(\ddagger)_{\text{loc}}$}
\def\ddownarrow{d\mskip2mu^{\downarrow}}
%\def\DiniD{\setover{
%  \hbox{\hskip.1em\vrule height-1pt width.5em depth1.4pt\hfil}}D}
%\def\Dinid{\hskip.1em
%  \hbox{\hskip.05em\vrule height-1.3pt width.5em depth1.7pt}\hskip-.65em D}
\def\Dinid{\hbox{$D$\hskip-8.3pt\vrule height-1pt width7pt depth1.85pt}
   \hskip1pt}
\def\DiniD{\hbox{$D$\hskip-8.3pt\vrule height9.2pt width7ptdepth-8.35pt}
   \hskip1pt}

\def\discrcenter#1#2{\ifdim\pagewidth>#1{#2}\else{

  \Centerline{#2}

  \noindent}\fi}
    %note:  in \discrcenter, make sure that #2 ends in a space
    % (if you want one to appear) and that the continuation
    % follows without
    % a space after the final } of #2, so that it will be
    % truly unindented
\def\discrpage{\ifnum\stylenumber=1{\frnewpage}  %redefined in mt.tex
        \else\ifnum\stylenumber=3{\frnewpage}
        \else\ifnum\stylenumber=5{\frnewpage}
        \else\wheader{}{18}{14}{2}{72pt}\fi\fi\fi}
\def\displaycause#1{\noalign{\noindent (#1)}}
\def\Displaycause#1{\cr\displaycause{#1}&}
\def\dotproduct{\hskip.1em\tbf{.}\hskip.1em}
\def\doubleheadrightarrow{\rightarrow\mskip-12mu\rightarrow}
\long\def\doubleinset#1{\inset{\inset{\parindent=-20pt #1}}}
\def\duparrow{d\mskip2mu^{\uparrow}}
\def\dvAformerly#1{\discrversionA{\footnote{Formerly #1.}}{}}
\def\dvAnew#1{\discrversionA{\footnote{New #1.}}{}}
\def\dvArevised#1{\discrversionA{\footnote{Revised #1.}}{}}
\def\dvrocolon#1{\dvro{#1: }{#1}}
\def\eightVVdash{\mskip5mu\vrule height 6pt depth 2pt width 0.4pt
  \mskip2mu\vrule height 6pt depth 2pt width 0.4pt
  \vrule height 2.25pt depth -1.8pt width 3.2pt\mskip2mu}
\def\embedsinto{\mskip5mu\raise1.5pt\hbox{$\subset$}\mskip-15mu
   \lower3.5pt\hbox{$\rightarrow$}\mskip5mu}
\def\energy{\mathop{\text{energy}}\nolimits}
%\eqalign:  see \S478 for breaks during \eqalign
\def\equivGT{\equiv_{\text{GT}}}
\def\equivT{\equiv_{\text{T}}}
\def\Eta{\text{H}}
\def\Expn{\Bbb E}
\long\def\Extract{\oldfootnote{}{
   \noindent{\eightit
       \ifresultsonly{Extract from {\eightsmc Measure Theory,}
          results-only version,
          by {\eightsmc D.H.Fremlin,}  University of Essex,
Colchester.}
       \else\ifwithproofs{Extract from {\eightsmc Measure Theory,}
          by {\eightsmc D.H.Fremlin,}  University of Essex,
Colchester.}
       \else{Extract from {\eightsmc Measure Theory,} abridged version,
          by {\eightsmc D.H.Fremlin,}  University of Essex,
Colchester.}
       \fi\fi
   This material is copyright.   It is issued under the terms of the
   Design Science License as published in
   {\eighttt http://dsl.org/copyleft/dsl.txt}.
   This is a development version and the
   source files are not permanently archived, but current versions are
normally
   accessible through
   {\eighttt https://www1.essex.ac.uk/maths/people/fremlin/mt.htm}.
   For further information contact {\eighttt david\@fremlin.org}.}
   }}
\def\Fn{\mathop{\text{Fn}}\nolimits}
\def\FN{\mathop{\text{FN}}\nolimits}
\def\formerly#1{\footnote{Formerly #1.}}
  %use only once in a chapter, never in same volume as reference
\def\fraction#1{\mathchoice{\hbox{$<$}#1\hbox{$>$}}
     {\hbox{$<$}#1\hbox{$>$}}
     {\hbox{$\scriptstyle<$}#1\hbox{$\scriptstyle>$}}
     {\hbox{$\scriptscriptstyle<$}#1\hbox{$\scriptscriptstyle>$}}}
\def\frakmctbl{\frak m_{\text{countable}}}
\def\frmedcirc{{\raise 2.5pt\hbox{$_{\bigcirc}$}}}
\def\grad{\mathop{\text{grad}}\nolimits}
\def\hatpmb#1{\hat{\pmb{#1}}}
\def\hattilde#1{\skew2\hat{\tilde#1}}
\def\hattildepmb#1{\skew1\hat{\tildepmb#1}}
\def\hcov{\mathop{\text{hcov}}\nolimits}
\def\Heint{\mathchoice{\hbox to 0pt{\hskip 3.1pt H\hss}}
     {\raise 0.6pt\hbox to 0pt{\hskip 1.9pt{\sevenrm H}\hss}}
     {\raise 0.4pt\hbox to 0pt{\hskip 1.5pt{\fiverm H}\hss}}
     {\raise 0.4pt\hbox to 0pt{\hskip 1.5pt{\fiverm H}\hss}}
     \intop\nolimits}
\def\high#1{\lceil#1\rceil}
\def\highoverline#1{\overline{\vrule height6pt width0pt depth 0pt#1}}
\def\Hint#1{({\it Hint\/}: #1)}
\def\hL{\mathop{\text{hL}}}
\def\Hmp{\text{Hom}_{\bar\mu,\pi}}
\def\hp{\mathop{\text{hp}}\nolimits}
\def\hwtr{hereditarily weakly $\theta$-{\vthsp}refinable}
\def\iiv{ii_{\pmb{v}}}
\def\iivu{\iiv(\pmb{u})}
\def\imp{inverse-{\vthsp}measure-{\vthsp}preserving}
\def\innerprod#1#2{(#1|#2)}
\def\intstar{\text{int*}}
\def\CalIS{\Cal I(\Cal S)}
\def\jump{\mathop{\text{jump}}\nolimits}
\def\Jump{\mathop{\text{Jump}}\nolimits}
\def\Krein{Kre\v{\i}n}
\def\Lambdapv{\Lambda_{\text{pv}}}
\def\leRB{\le_{\text{RB}}}
\def\leRK{\le_{\text{RK}}}
\def\link{\mathord{\text{link}}}
\def\locmafp{\widehat{\otimes}_{\text{loc}}}
\def\Locmafp{\widehat{\bigotimes}^{\text{loc}}}
\def\low#1{\lfloor#1\rfloor}
\def\lti{left-\vthsp translation-\vthsp invariant}
\def\magnitude{\mathop{\text{mag}}}
\def\Mah{\mathop{\text{Mah}}}
\def\Mahcr{\Mah_{\text{cr}}}
\def\MahcrR{\Mah_{\text{crR}}}
\def\MahqR{\Mah_{\text{qR}}}
\def\MahR{\Mah_{\text{R}}}
\def\MahspcrR{\Mah_{\text{spcrR}}}
\def\Malm{M_{\text{alm}}}
\def\Marik{Ma\v r\'\i k} 
\def\Mbv{M_{\text{bv}}}
\def\Mcdlg{M_{\text{cdlg}}}
\def\Mdm{M_{\text{D-M}}}
\def\med{\mathop{\text{med}}\nolimits}
\def\Mfa{M_{\text{fa}}}
\def\Mhsr{\mathop{\text{Mhsr}}\nolimits}
\def\Mjf{\mathop{M_{\text{j-f}}}\nolimits}
\def\Mlbv{M_{\text{lbv}}}
\def\Mlob{M_{\text{lo-b}}}
\def\Mlns{M_{\text{ln-s}}}
\def\Mmo{M_{\text{mo}}}
\def\Mlmo{M_{\text{lmo}}}
\def\Mns{M_{\text{n-s}}}
\def\Mnsu{M_{\text{n-s}}^{\uparrow}}
%\def\Mnsu{M_{\text{n-s}\uparrow}}
\def\MMah{M_{\text{Mah}}}
\def\Mob{M_{\text{o-b}}}
\def\Mpob{M_{\text{po-b}}}
\def\Mpv{M_{\text{pv}}}
\def\MSi{M_{\text{S-i}}}
\def\Msimp{M_{\text{simp}}}
\def\Mth{Ma\-ha\-ram-{\vthsp}type-{\vthsp}homogeneous}
\def\Mvlm{M_{\text{vlm}}}
\def\negquad{\hskip-1em\relax}
\def\NN{\BbbN^{\Bbb N}}
\def\non{\mathop{\text{non}}}
\def\notBooleanop#1{
   \ifamstexsupported{\mathchoice
{\hbox{\hskip.2em$\not\scriptfont2=\eightsy\scriptstyle#1$\hskip.2em}}
{\hbox{\hskip.2em$\not\scriptfont2=\eightsy\scriptstyle#1$\hskip.2em}}
    {{\not\scriptstyle#1}}
    {{\not\scriptscriptstyle#1}}}
   \else{\mathchoice{{\not\scriptstyle#1}}{{\not\scriptstyle#1}}
    {{\not\scriptstyle#1}}{{\not\scriptscriptstyle#1}}}
    \fi}
\def\notBsubseteq{\notBooleanop{\subseteq}}
\def\notBsupseteq{\notBooleanop{\supseteq}}
\def\notVVdash{\not\mskip4mu\vrule height 7.5pt depth 2.5pt width 0.5pt
  \mskip2.5mu\vrule height 7.5pt depth 2.5pt width 0.5pt
  \vrule height 2.75pt depth -2.25pt width 4pt\mskip-2mu}
\def\nuprime{\nu^{\mskip0.7mu\prime}} %better  \nu'  in superscripts
\def\nw{\mathop{\text{nw}}}
\def\Osclln{\mathop{\text{Osclln}}\nolimits}
\def\otp{\mathop{\text{otp}}}
\def\partstar{\mathchoice{\partial\hbox{*}}
     {\partial\hbox{*}}
     {\partial\hbox{\sevenrm *}}
     {\partial\hbox{\fiverm *}}}
\def\per{\mathop{\text{per}}}
\def\Pfint{\mathchoice{\hbox to 0pt{\hskip 3.1pt Pf\hss}}
     {\raise 0.6pt\hbox to 0pt{\hskip 1.3pt{\sevenrm Pf}\hss}}
     {\raise 0.4pt\hbox to 0pt{\hskip 1.0pt{\fiverm Pf}\hss}}
     {\raise 0.4pt\hbox to 0pt{\hskip 1.0pt{\fiverm Pf}\hss}}
     \intop\nolimits
     \mathchoice{\hskip 2pt}{\hskip 1.2pt}{\hskip 1pt}{\hskip 1pt}}
\def\Pou{\mathop{\text{Pou}}\nolimits}
\def\prforc{\preccurlyeq_{\text{forcing}}}
\def\prGT{\preccurlyeq_{\text{GT}}}
\def\prT{\preccurlyeq_{\text{T}}}
\def\pssqa{power set $\sigma$-quotient algebra}
\def\qm{quasi-{\vthsp}measurable}
\def\Reverse#1{\setover
   {\lower0.4ex\hbox{$\scriptscriptstyle\leftrightarrow$}}{#1}}
\def\rhoKR{\rho_{\text{KR}}} 
\def\rhotv{\rho_{\text{tv}}} 
\def\Rint{\mathchoice{\hbox to 0pt{\hskip 3.1pt R\hss}}
     {\raise 0.6pt\hbox to 0pt{\hskip 1.9pt{\sevenrm R}\hss}}
     {\raise 0.4pt\hbox to 0pt{\hskip 1.5pt{\fiverm R}\hss}}
     {\raise 0.4pt\hbox to 0pt{\hskip 1.5pt{\fiverm R}\hss}}
     \intop\nolimits}
\def\RO{\text{RO}}
\def\rti{right-{\vthsp}translation-{\vthsp}invariant}
\def\rvm{real-{\vthsp}valued-{\vthsp}measurable}
\def\sat{\mathop{\text{sat}}}
\def\sfamilysS#1{\family{\sigma}{\Cal S}{#1_{\sigma}}}
\def\sfcc{$\sigma$-finite-cc}
\def\sgn{\mathop{\text{sgn}}}
\def\shr{\mathop{\text{shr}}\nolimits}
\def\Sii{\text{S}ii}
\def\Siiv{\Sii_{\pmb v}}
\def\Sint{\mathchoice{\hbox to 0pt{\hskip 4.1pt\ninerm S\hss}} 
     {\raise 0.6pt\hbox to 0pt{\hskip 3.0pt{\sevenrm S}\hss}}
     {\raise 0.4pt\hbox to 0pt{\hskip 1.5pt{\fiverm S}\hss}} %unchecked
     {\raise 0.4pt\hbox to 0pt{\hskip 1.5pt{\fiverm S}\hss}} %unchecked
     \intop\nolimits}
\def\Sinteightpoint{\raise 0.6pt\hbox to 0pt{\hskip 2.3pt{\fiverm S}\hss}
     \intop\nolimits}
\def\SL{\text{SL}}
\def\sm{semi-martingale} %hyphen to remind me that it's the "virtual" kind
\def\spread{\mathop{\text{spr}}}
\def\ssplrarrow{^{\scriptscriptstyle\leftrightarrow}}
\def\ssptilde{{\raise1.2ex\hbox{$\scriptscriptstyle\sim$}}}
%\def\sspvec{{\raise1.2ex\hbox{$\scriptscriptstyle\rightarrow$}}}
\def\sspvec{\mskip3mu\vec{\null}\mskip5mu}
\def\Sti{\mathop{\text{Sti}}\nolimits}
\def\strbar{^{\scriptscriptstyle\pmb{-}}}
\def\strplus{^{\scriptscriptstyle\pmb{+}}}
\def\strprime{^{\mathchoice{\mskip1.5mu\pmb{\prime}}
                  {\mskip1.5mu\pmb{\prime}}
                  {\mskip1.5mu\pmb{\prime}}
                  {\mskip1mu\pmb{\prime}}}}
%\def\strprime{\,\pmb{\prime}}
\def\supp{\mathop{\text{supp}}}
\def\tausm{\tau_{\sigma\text{-m}}}
\def\tbfoneminus#1{\rlap{$\tbf 1^{(#1)}$}\rlap{\hskip7pt$_<$}
   \phantom{\tbf 1^{(#1)}}}
%\def\tbfoneminus#1{\rlap{$\tbf 1^{(#1)}$}\rlap{\hskip7pt$_-$}
%   \phantom{\tbf 1^{(#1)}}}
\def\tensorhat{\widehat{\otimes}}
\def\Tensorhat{\widehat{\bigotimes}}
\def\tensorhatsigma{\widehat{\otimes}_{\sigma}}
\def\tildeClll{\hbox{$\tilde C$
  \hskip-4.5pt\raise4pt\hbox{\char'23
  \hskip-1.6pt\vrule height6.5pt width.5pt depth0pt
  \hskip.7pt\vrule height6.5pt width.5pt depth0pt
  \hskip-2.5pt\lower5.4pt\hbox{\char'23}}}}
\def\tildehat#1{\skew2\tilde{\hat#1}}
\def\tildehatpmb#1{\tilde{\hatpmb#1}}
\def\tildepmb#1{\tilde{\pmb{#1}}}
\def\Tr{\mathop{\text{Tr}}}
\def\trs{^{\top}\hskip-1pt}
\def\triplepc#1#2#3{(#1,#2,\hbox{$<$}#3)}
\def\TSi{\frak T_{\text{S-i}}}
\def\undphi{\underline{\phi}{}}
\def\undpsi{\underline{\psi}{}}
\def\undtheta{\underline{\theta}{}}
\def\updown{\mathchoice
  {\hbox{\hskip-1.5pt\char'23
    \hskip-1.7pt\vrule height6.5pt width.7pt depth1.3pt
    \hskip.9pt\vrule height6.5pt width.7pt depth1.3pt
    \hskip-3.3pt\lower6.7pt\hbox{\char'23}\hskip-1.4pt}}
  {\hbox{\hskip-2pt\char'23
    \hskip-1.7pt\vrule height6.5pt width.7pt depth1.3pt
    \hskip.9pt\vrule height6.5pt width.7pt depth1.3pt
    \hskip-2.5pt\lower6.7pt\hbox{\char'23}\hskip-1.4pt}}
  {\hbox{\hskip-2pt\char'23  %OK for subscript, fails for superscript
    \hskip-1.7pt\vrule height6.5pt width.7pt depth0pt
    \hskip.9pt\vrule height6.5pt width.7pt depth0pt
    \hskip-2.5pt\lower5.4pt\hbox{\char'23}\hskip-1.4pt}}
  {\hbox{\hskip-2pt\char'23  %not tested
    \hskip-1.7pt\vrule height6.5pt width.7pt depth1.3pt
    \hskip.9pt\vrule height6.5pt width.7pt depth1.3pt
    \hskip-2.5pt\lower6.7pt\hbox{\char'23}\hskip-1.4pt}}}
\def\updownarrows{\uparrow\mskip-4mu\downarrow}
\def\updowncgt{$\updown$-convergent}
\def\upr{\mathop{\text{upr}}\nolimits}
\def\ureg{\mathop{\text{ureg}}}

\def\Var{\mathop{\text{Var}}\nolimits}

\def\varaction{\mathchoice{\lower-.2ex\hbox{$\smallcirc$}}
                       {\lower-.2ex\hbox{$\smallcirc$}}
                       {\ssbullet}
                       {\ssbullet}}
\def\varcheck#1{{\setover
       {\lower 0.4ex\hbox{$\scriptscriptstyle\vee$}}{#1}}}
\def\varcheckf{{\setover{\,\lower 0.4ex
              \hbox{$\scriptscriptstyle\vee$}}{f}}}
\def\varchecki{\mskip-3mu\text{\it{\accent"14{\hbox{\hskip-3.5pt\i}}}}\mskip2mu}
\def\varspcheck{^{\scriptscriptstyle\vee}}
%\def\varspcheck{{\setover{\scriptscriptstyle\vee}{\phantom{g}}}}
%\def\varspcheckf{{\setover{\lower 0.4ex
%              \hbox{$\scriptscriptstyle\vee$}}{\phantom{f}}}}
\def\varhat#1{{\setover{\lower 0.4ex
              \hbox{$\scriptscriptstyle\wedge$}}{#1}}}
\def\varhatf{{\setover{\,\lower 0.4ex
              \hbox{$\scriptscriptstyle\wedge$}}{f}}}
\def\varinnerprod#1#2{#1\dotproduct#2}
\def\varsphat{^{\scriptscriptstyle\wedge}}
% or use  \hbox{\^{\null}}
%\def\varsphat{{\setover{\scriptscriptstyle\wedge}{\phantom{g}}}}
%\def\varsphatf{{\setover{\lower 0.4ex
%              \hbox{$\scriptscriptstyle\wedge$}}{\phantom{f}}}}
\def\varBspcheck{\mathchoice
  {\hskip-2pt
  {\fourteenrm\hbox{\lower2pt\hbox{\v{\null}}}}\hskip-1pt}
  {\hskip-2pt
  {\fourteenrm\hbox{\lower2pt\hbox{\v{\null}}}}\hskip-1pt}
  {\hskip-2.5pt
  {\fourteenrm\hbox{\lower4pt\hbox{\v{\null}}}}\hskip-1pt}
  {\hskip-2.5pt
  {\fourteenrm\hbox{\lower4pt\hbox{\v{\null}}}}\hskip-1pt}}
\def\varCspcheck{\mathchoice
  {\hskip-1pt\check{\phantom x}}   %not tested
  {\hskip-1pt\check{\phantom x}}
  {\hskip-1.2pt\check{\phantom x}}
  {\hskip-1.5pt\check{\phantom x}}}  %not tested
\def\varCsphat{\mathchoice
  {\hskip-1pt\hat{\phantom x}}   %not tested
  {\hskip-1pt\hat{\phantom x}}
  {\hskip-1.2pt\hat{\phantom x}}  %not tested
  {\hskip-1.5pt\hat{\phantom x}}}  %not tested
\def\VeqL{$V\hbox{$=$}L$}
\def\VVdash{\mskip5mu\vrule height 7.5pt depth 2.5pt width 0.5pt
  \mskip2.5mu\vrule height 7.5pt depth 2.5pt width 0.5pt
  \vrule height 2.75pt depth -2.25pt width 4pt\mskip2mu}
\def\VVdP{\VVdash_{\Bbb P}}
\def\VVdPk{\VVdash_{\Bbb P_{\kappa}}}
\def\wdistr{\mathop{\text{wdistr}}}
\def\weight{\mathop{\text{wt}}}
\def\w1lim{\mathop{\text{w1lim}}}
\def\wsid{weakly $(\sigma,\infty)$-{\vthsp}distributive}
\def\1lim{\mathop{1\mskip-1.5mu\text{lim}}}
\def\2vm{two-{\vthsp}valued-{\vthsp}measurable}

%Cara\-th\'e\-o\-dory

\def\chaptername{}
\def\sectionname{}
\def\headlinesectionname{}

\loadeusm\loadeurm
\Loadfourteens
\Loadfont{\ninerm=cmr9}

%note:  $..\cmmnt{=..}..$  is better as  \cmmnt{\mskip5mu
 
  
%----------------------------------------------------------%  
  
\smallprint  
%\USsmallprint  
%\noproofs  
%\resultsonly  
%\atUEssex  
\usegraphicx
  
\Extract  
  
%----------------------------------------------------------%  

\frfilename{mt65.tex}
\versiondate{31.10.17}
\copyrightdate{2014}

\def\chaptername{Applications}

\newchapter{65}

At long last I turn again to some of the results for which the theory
outlined in this volume was
developed.   I start with some relatively elementary ideas using nothing
more advanced than \S624, showing that locally jump-free virtual local
martingales are associated with `exponential' processes of the same kind
(651C).   These in turn are associated with identities for integral
equations (651G, 651K) and change-of-law results (651I).
Ideas at the same level take us to L\'evy's characterization of Brownian
motion (653F);  going deeper, and using the time-changes of \S635, we
can represent many locally jump-free local martingales in terms of
Brownian motion (653G).

The exponential processes of \S651 can be thought of as solutions of a
particularly simple kind of stochastic differential equation.   Working
very much harder, we find that we have versions of Picard's theorem,
for integral equations with a Lipschitz condition on the integrand,
for both the Riemann-sum integral (654I) and the S-integral (654N).
A twist in the theory of exponential processes, with a refinement inspired
by the theory of financial markets, leads us to the famous Black-Scholes
equation (655D).

\discrpage

\frfilename{mt651.tex}
\versiondate{11.9.23}

\def\chaptername{Applications}
\def\sectionname{Exponential processes}

\newsection{651}

Associated with any jump-free integrator is an `exponential process'
(651B);  if the integrator is a martingale, the exponential process may be
a uniformly integrable martingale (651D-651E).   
This gives us an important class of
non-negative martingales which we can use in change-of-law results (651J).

\leader{651A}{Notation}\cmmnt{ As always, \Asissix\ is a stochastic
integration structure.
$L^0=L^0(\frak A)$;  if $h:\Bbb R\to\Bbb R$ is a Borel measurable function,
I write $\bar h$ for either of the corresponding operators on members of $L^0$
(612Ac) and on processes in $L^0$ (612B).   Limits in $L^0$ will be taken with
respect to the topology of convergence in measure (613B).
% unless otherwise indicated.   
For a fully adapted process $\pmb u=\sfamilysS{u}$,
$u_{\downarrow}=\lim_{\sigma\downarrow\Cal S}u_{\sigma}$ is its
starting value if this is defined (613Bk).   For local integrators
$\pmb v$ and $\pmb w$, $\covar{\pmb v}{\pmb w}$ is their covariation
and $\pmb v^*$ is the quadratic variation of $\pmb v$ (617H).

$\Expn=\Expn_{\bar\mu}$ is integration with respect to
$\bar\mu$\cmmnt{ as in \S\S365 and 613,} and
$L^1_{\bar\mu}\cmmnt{\mskip5mu=L^1(\frak A,\bar\mu)}$ is the
corresponding $L$-space, while $\theta(w)=\Expn(|w|\wedge\chi 1)$ for
$w\in L^0$ (613Ba).   For $\tau\in\Cal T$,
$P_{\tau}:L^1_{\bar\mu}\to L^1_{\bar\mu}\cap L^0(\frak A_{\tau})$ is the
conditional expectation operator associated with the closed subalgebra
$\frak A_{\tau}$.
For $z\in L^1_{\bar\mu}$, $\pmb{P}z$ is the martingale
$\family{\tau}{\Cal T}{P_{\tau}z}$.
For $y\in L^0(\frak A)$, $y\tbf{1}$ is the
constant process on $\{\tau:y\in L^0(\frak A_{\tau})\}$
with value $y$ (612De).}

\leader{651B}{Theorem} Let $\Cal S$ be a non-empty sublattice of $\Cal T$.
Suppose that $\pmb v$ is a locally
jump-free local integrator, and $\pmb u$ a locally moderately oscillatory
process, both with domain $\Cal S$.   Set
$\pmb z=\highoverline{\exp}(\pmb v-v_{\downarrow}\tbf{1}-\bover12\pmb v^*)$.
Then $\pmb z$ is a locally jump-free local integrator,
$\pmb z=\tbf 1+\iiv(\pmb z)$ and
$ii_{\pmb z}(\pmb u)=\iiv(\pmb u\times\pmb z)$.
If $\pmb v$ is in fact a
jump-free integrator and $\pmb u$ a moderately oscillatory process,
then $\pmb z$ is a jump-free integrator

\proof{{\bf (a)} Suppose, to begin with, that 
$\pmb v=\family{\sigma}{\Cal S}{v_{\sigma}}$ is a jump-free
integrator and $\pmb u$ is moderately oscillatory.

\medskip

\quad{\bf (i)} By 616Ib and 615Gb, $v_{\downarrow}$ is defined.
By 617H-617I, $\pmb v^*$ is defined everywhere on $\Cal S$ and is an integrator;
by 618T, it is jump-free;  by 618Ga,
$\pmb w=\pmb v-v_{\downarrow}\tbf{1}-\bover12\pmb v^*$ and
$\pmb z=\highoverline{\exp}(\pmb w)$ are jump-free, therefore moderately
oscillatory.

Express $\pmb v^*$, $\pmb w$ and $\pmb z$ as $\sfamilysS{v^*}$,
$\sfamilysS{w}$ and $\sfamilysS{z}$.   We have
$w_{\downarrow}=\lim_{\sigma\downarrow\Cal S}v^*_{\sigma}=0$ (617J(b-i)).

Because $\pmb v$ and $\pmb v^*$ are integrators,
so is $\pmb w$;  by 616N, $\pmb z$ is an integrator.
Now $v_{\downarrow}\tbf{1}+\bover12\pmb v^*$ is a jump-free integrator of
bounded variation, so

\Centerline{$\pmb w^*
=\pmb v^*-2\covar{\pmb v}{v_{\downarrow}\tbf 1+\bover12\pmb v^*}
  +(v_{\downarrow}\tbf 1+\bover12\pmb v^*)^*
=\pmb v^*$}

\noindent by 624C.

\medskip

\quad{\bf (ii)} We have

$$\eqalignno{\int_{\Cal S}\pmb u\,d\pmb z&
=\int_{\Cal S}\pmb u\,d(\highoverline{\exp}(\pmb w))\cr&
=\int_{\Cal S}\pmb u\times\highoverline{\exp}(\pmb w)\,d\pmb w
   +\Bover12\int_{\Cal S}\pmb u\times\highoverline{\exp}(\pmb w)\,d\pmb w^*
\Displaycause{619C, because $\exp'=\exp''=\exp$}
=\int_{\Cal S}\pmb u\times\pmb z\,d\pmb w
   +\Bover12\int_{\Cal S}\pmb u\times\pmb z\,d\pmb v^*
\Displaycause{because $\pmb w^*=\pmb v^*$}
=\int_{\Cal S}\pmb u\times\pmb z\,
    d(\pmb w+v_{\downarrow}\tbf 1+\bsover12\pmb v^*)
\Displaycause{$\int_{\Cal S}\pmb u\times\pmb z\,d(v_{\downarrow}\tbf{1})=0$
because $v_{\downarrow}\Delta\tbf{1}$ is the zero interval function}
=\int_{\Cal S}\pmb u\times\pmb z\,d\pmb v.\cr}$$

\medskip

{\bf (b)} In the general case, in which $\pmb v$ is a locally jump-free local
integrator and $\pmb u$ is locally moderately oscillatory, then for any
$\tau\in\Cal S$ we see that $\pmb z\restr\Cal S\wedge\tau$ is a jump-free
integrator and $\int_{\Cal S\wedge\tau}\pmb u\,d\pmb z
=\int_{\Cal S\wedge\tau}\pmb u\times\pmb z\,d\pmb v$.   So $\pmb z$ is a
locally jump-free local integrator and
$ii_{\pmb z}(\pmb u)=\iiv(\pmb u\times\pmb z)$.   Now

\Centerline{$\pmb z
=z_{\downarrow}\tbf 1+ii_{\pmb z}(\tbf 1)=\tbf 1+\iiv(\pmb z)$}

\noindent because

\Centerline{$z_{\downarrow}=\highoverline{\exp}(w_{\downarrow})
=\highoverline{\exp}(0)=\chi 1$.}

}%end of proof of 651B

\leader{651C}{Corollary} Let $\Cal S$ be a
non-empty sublattice of $\Cal T$, and
$\pmb v$ a locally jump-free virtually local
martingale with domain $\Cal S$.   Then
$\pmb z=\highoverline{\exp}(\pmb v-v_{\downarrow}\tbf{1}-\bover12\pmb v^*)$
is a locally jump-free virtually local martingale.
%"Girsanov's theorem" in Bain 09, \S14 and Wikipedia, which lumps 651J in too

\proof{ By 623Kd, $\pmb v$ is a local integrator, so 651B tells us that
$\pmb z$ is a locally
jump-free local integrator (therefore locally moderately oscillatory)
and $\pmb z=\tbf 1+\iiv(\pmb z)$.   By 623O,
$\iiv(\pmb z)$ and therefore $\pmb z$ are virtually local martingales.
}% end of proof of 651C

\leader{651D}{}\cmmnt{ It is useful to know when
$\highoverline{\exp}(\pmb v-v_{\downarrow}\tbf{1}-\bover12\pmb v^*)$
is actually a martingale.
The following criterion gives us a sufficient condition.

\medskip

\noindent}{\bf Theorem} Let $\Cal S$ be a non-empty finitely full sublattice of
$\Cal T$ and
$\pmb v=\family{\sigma}{\Cal S}{v_{\sigma}}$ a
locally jump-free virtually local martingale\cmmnt{ with quadratic variation
$\pmb v^*=\family{\sigma}{\Cal S}{v^*_{\sigma}}$}.   Let
$\pmb z=\highoverline{\exp}(\pmb v-v_{\downarrow}\tbf{1}-\bover12\pmb v^*)$
be the associated exponential process\cmmnt{, as in 651B-651C}.
If $\cmmnt{\beta=\mskip5mu}\sup_{\sigma\in\Cal S}
\Expn(\highoverline{\exp}(\bover12(v_{\sigma}-v_{\downarrow})))$ is finite,
then $\pmb z$ is a uniformly integrable martingale.

%"Kazamaki's criterion" in Protter p 141, Kazamaki 77 ?

\proof{{\bf (a)} Since $\pmb v-v_{\downarrow}\tbf{1}$ is a locally
jump-free virtually local martingale (using 623Kg) 
with quadratic variation $\pmb v^*$,
it is enough to consider the case $v_{\downarrow}=0$, so that
$\pmb z=\highoverline{\exp}(\pmb v-\bover12\pmb v^*)$.   Being a
virtually local martingale (651C) and non-negative,
$\pmb z$ is a $\|\,\|_1$-bounded supermartingale (627Da)
and, setting $z_{\sigma}=\highoverline{\exp}(v_{\sigma}-\bover12v^*_{\sigma})$,

\Centerline{$|\|z_{\sigma}\|_1=\Expn(z_{\sigma})\le\Expn(z_{\downarrow})=1$}

\noindent for every $\sigma\in\Cal S$.
As $\Cal S$ is finitely full, $\pmb z$ is an approximately local martingale
(623K(b-iii)), so $w=\lim_{\sigma\uparrow\Cal S}z_{\sigma}$ is defined (623La), 
and $\Expn(w)=\|w\|_1\le 1$ (613Bc). 

\medskip

{\bf (b)} The next thing to observe is that $\pmb v$ is
$\|\,\|_1$-bounded.    \Prf\ Again because $\Cal S$ is finitely full, $\pmb v$
also is an approximately local martingale.   
Take $\tau\in\Cal S$ and $\epsilon>0$.
Then there is a non-empty downwards-directed subset $A\subseteq\Cal S$ such that
$\sup_{\rho\in A}\bar\mu\Bvalue{\rho<\tau}\le\epsilon$ and
$R_A(\pmb v)$, as defined in 623B, is a martingale.   Express it as 
$\family{\sigma}{\Cal S}{v_{A\sigma}}$.   Then 
$\lim_{\sigma\downarrow\Cal S}v_{A\sigma}=v_{\downarrow}=0$ (623B(c-i)).
By 622Ed, $0$ is the $\|\,\|_1$-limit $\1lim_{\sigma\downarrow\Cal S}v_{A\sigma}$
(622Ed) and 
$\Expn(v_{A\tau})=\lim_{\sigma\downarrow\Cal S}\Expn(v_{A\sigma})=0$.   
It follows that 

\Centerline{$\|v_{A\tau}\|_1=\Expn(v_{A\tau}^+)+\Expn(v_{A\tau}^-)
=2\Expn(v_{A\tau}^+)=2\|v_{A\tau}^+\|_1$.}

By definition, $v_{A\tau}=\lim_{\rho\downarrow A}v_{\tau\wedge\rho}$, so
$v_{A\tau}^+=\lim_{\rho\downarrow A}v_{\tau\wedge\rho}^+$ (since
$x\mapsto x^+:L^0\to L^0$ is continuous), while 
$x^+\le 2\,\highoverline{\exp}(\bover12x)$ for any $x\in L^0$
because
$\max(\alpha,0)\le 2\exp(\bover12\alpha)$ for every $\alpha\in\Bbb R$.
So 

\Centerline{$\|v_{\tau\wedge\rho}^+\|_1=\Expn(v_{\tau\wedge\rho}^+)
\le 2\Expn(\highoverline{\exp}(\bover12v_{\tau\wedge\rho})\le 2\beta$}

\noindent for every $\rho\in A$ and 

\Centerline{$\|v_{A\tau}\|_1=2\|v_{A\tau}^+\|_1
\le 2\sup_{\rho\in A}\|v_{\tau\wedge\rho}^+\|_1\le 4\beta$,}

\noindent while 

\Centerline{$\theta(v_{\tau}-v_{A\tau})\le\bar\mu\Bvalue{v_{\tau}\ne v_{A\tau}}
\le\bar\mu(\sup_{\rho\in A}\Bvalue{v_{\tau}\ne v_{\tau\wedge\rho}})
\le\bar\mu(\sup_{\rho\in A}\Bvalue{\rho<\tau})\le\epsilon$.}

\noindent As $\epsilon$ is arbitrary, $\|v_{\tau}\|_1\le 4\beta$;  as
$\tau$ is arbitrary. $\pmb v$ is $\|\,\|_1$-bounded.\ \Qed

By 623La again, it follows that we have a
limit $v=\lim_{\sigma\uparrow\Cal S}v_{\sigma}$.   In this case,
$\highoverline{\exp}(\bover12v)
=\lim_{\sigma\uparrow\Cal S}\highoverline{\exp}(\bover12v_{\sigma})$ (613Bb),
so

\Centerline{$\Expn(\highoverline{\exp}(\bover12v))
\le\liminf_{\sigma\uparrow\Cal S}
  \Expn(\highoverline{\exp}(\bover12v_{\sigma}))\le\beta$}

\noindent (613Bc once more).

\medskip

{\bf (c)} For $0<\alpha<1$ and $\sigma\in\Cal S$ set
$z_{\alpha\sigma}
=\highoverline{\exp}(\alpha v_{\sigma}-\bover12\alpha^2v^*_{\sigma})$.
If $q=\Bover1{2\alpha-\alpha^2}$, then $q>1$ and

\Centerline{$\Expn(z_{\alpha\sigma}^q)
\le\beta^{(2-2\alpha)/(2-\alpha)}$.}

\noindent\Prf\

$$\eqalign{z_{\alpha\sigma}^q
&=\highoverline{\exp}(q\alpha v_{\sigma}
   -\bsover12q\alpha^2v^*_{\sigma})
=\highoverline{\exp}(\bsover1{2-\alpha}v_{\sigma}
   -\bsover{\alpha}{2(2-\alpha)}v^*_{\sigma})\cr
&=\highoverline{\exp}(\bsover{\alpha}{2-\alpha}v_{\sigma}
      -\bsover{\alpha}{2(2-\alpha)}v^*_{\sigma})
   \times\highoverline{\exp}
       (\bsover{1-\alpha}{2-\alpha}v_{\sigma})\cr}$$

\noindent so, by H\"older's inequality (244Eb),

$$\eqalignno{\Expn(z_{\alpha\sigma}^q)
&\le\|\highoverline{\exp}(\bsover{\alpha}{2-\alpha}v_{\sigma}
   -\bsover{\alpha}{2(2-\alpha)}v^*_{\sigma})\|_{(2-\alpha)/\alpha}
   \|\highoverline{\exp}(\bsover{1-\alpha}{2-\alpha}v_{\sigma})
      \|_{(2-\alpha)/(2-2\alpha)}\cr
&=\bigl(\Expn(\highoverline{\exp}(v_{\sigma}
   -\bsover12v^*_{\sigma}))\bigr)^{\alpha/(2-\alpha)}\,
   \bigl(\Expn(\highoverline{\exp}(\bsover12v_{\sigma}))
       \bigr)^{(2-2\alpha)/(2-\alpha)}\cr
&\le\bigl(\Expn(\highoverline{\exp}(\bsover12v_{\sigma}))
         \bigr)^{(2-2\alpha)/(2-\alpha)}
\le\beta^{(2-2\alpha)/(2-\alpha)}.   \text{ \Qed}\cr}$$

It follows that $\{z_{\alpha\sigma}:\sigma\in\Cal S\}$ is uniformly
integrable (621Be).   But $\alpha\pmb v$ is a locally
jump-free virtually local martingale with quadratic variation
$\alpha^2\pmb v^*$, starting from $0$, so
$\family{\sigma}{\Cal S}{z_{\alpha\sigma}}$ is a virtually local martingale, by
651C;  as $\Cal S$ is finitely full, it is an approximately local martingale
(623K(b-iii) again);  by 623Nb, it is in fact a martingale, and
$w_{\alpha}=\lim_{\sigma\uparrow\Cal S}z_{\alpha\sigma}$ is defined, with

\Centerline{$\Expn(w_{\alpha})
=\lim_{\sigma\uparrow\Cal S}\Expn(z_{\alpha\sigma})
=\lim_{\sigma\downarrow\Cal S}\Expn(z_{\alpha\sigma})
=\Expn(\lim_{\sigma\downarrow\Cal S}z_{\alpha\sigma})
=1$.}

\medskip

{\bf (d)} For each $\alpha\in\ooint{0,1}$ and $\sigma\in\Cal S$,

$$\eqalign{z_{\alpha\sigma}
&=\highoverline{\exp}(\alpha v_{\sigma}-\bsover12\alpha^2v^*_{\sigma})
=\highoverline{\exp}(\alpha^2v_{\sigma}-\bsover12\alpha^2v^*_{\sigma})
  \times\highoverline{\exp}(\alpha(1-\alpha)v_{\sigma})\cr
&=z_{\sigma}^{\alpha^2}
  \times\highoverline{\exp}(\alpha(1-\alpha)v_{\sigma});\cr}$$

\noindent taking the limit as $\sigma\uparrow\Cal S$,

\Centerline{$w_{\alpha}
=w^{\alpha^2}\times\highoverline{\exp}(\alpha(1-\alpha)v)$}

\noindent (because multiplication and the operations
$u\mapsto|u|^{\alpha^2}$, $u\mapsto\highoverline{\exp}(\alpha(1-\alpha)u)$
from $L^0$ to itself are continuous for the topology of convergence in
measure).   By H\"older's inequality again,

$$\eqalign{1
&=\Expn(w_{\alpha})
\le\|w^{\alpha^2}\|_{1/\alpha^2}\,
       \|\highoverline{\exp}(\alpha(1-\alpha)v)\|_{1/(1-\alpha^2)}\cr
&=\bigl(\Expn(w)\bigr)^{\alpha^2}\,
       \bigl(\Expn((\highoverline{\exp}
          (\alpha(1-\alpha)v))^{1/(1-\alpha^2)})\bigr)^{1-\alpha^2}
	  \cr
&=\bigl(\Expn(w)\bigr)^{\alpha^2}\,
       \bigl(\Expn((\highoverline{\exp}
          (\bsover12v))^{2\alpha/(1+\alpha)})
	     \bigr)^{1-\alpha^2}.\cr}$$

\noindent Taking the limit as $\alpha\uparrow 1$, we get

\Centerline{$\|w\|_1=\Expn(w)\ge 1\ge\sup_{\sigma\in\Cal S}\|z_{\sigma}\|_1$}	
	
\noindent by (a).

\medskip

{\bf (e)} By 623N(b-iii), it follows that $\pmb z$ is a
uniformly integrable martingale.
}%end of proof of 651D

\leader{651E}{Corollary} Let $\Cal S$ be a non-empty sublattice of
$\Cal T$ and $\pmb v$ a locally jump-free virtually local martingale
with domain $\Cal S$.   Let
$\pmb z=\highoverline{\exp}(\pmb v-v_{\downarrow}\tbf{1}-\bover12\pmb v^*)$
be the associated exponential process.

(a) If $\highoverline{\exp}(\bover12\pmb v^*)$ is $\|\,\|_1$-bounded,
then $\pmb z$ is a uniformly integrable martingale.
%"Novikov's criterion" Protter p 141;  Novikov 72?

(b) If $\pmb v^*$ is an $L^{\infty}$-process, then
$\pmb z$ is a martingale.

\proof{{\bf (a)} As in 651D, it is enough to consider the case in which
$v_{\downarrow}=0$.   

\medskip

\quad{\bf (i)} Suppose to begin with that $\Cal S$ is finitely full.
Express $\pmb v$, $\pmb v^*$ and $\pmb z$ as
$\sfamilysS{v}$, $\sfamilysS{v^*}$ and $\sfamilysS{z}$.

For any non-negative $u$, $v\in L^0$,

\Centerline{$(\Expn(\sqrt{u\times v}))^2
\le\|\sqrt{u}\|_2^2\,\|\sqrt{v}\|_2^2
=\Expn(u)\,\Expn(v)$.}

\noindent Now for any $\sigma\in\Cal S$ we have
$\highoverline{\exp}(v_{\sigma})
=z_{\sigma}\times\highoverline{\exp}(\bover12v^*_{\sigma})$, so

\Centerline{$(\Expn(\highoverline{\exp}(\bsover12v_{\sigma})))^2
\le\Expn(z_{\sigma})\,\Expn(\highoverline{\exp}(\bsover12v^*_{\sigma}))
\le\Expn(\highoverline{\exp}(\bsover12v^*_{\sigma}))$}

\noindent because $\pmb z$ is a supermartingale
starting at $\chi 1$, as in part (a) of the proof of 651D.   Accordingly

\Centerline{$\sup_{\sigma\in\Cal S}
   \Expn(\highoverline{\exp}(\bover12v_{\sigma}))
\le\sqrt{\sup_{\sigma\in\Cal S}\Expn(\highoverline{\exp}(\bsover12v^*_{\sigma}))}$}

\noindent is finite, and $\pmb z$ is a uniformly integrable martingale, by
651D.

\medskip

\quad{\bf (ii)} For the general case, let $\hat\Cal S_f$ be the finitely-covered
envelope of $\Cal S$, and 
$\hatpmb v=\family{\tau}{\hat\Cal S_f}{\hat v_{\tau}}$, 
$\hatpmb v^*$ and $\hatpmb z$ the fully
adapted extensions of $\pmb v$, $\pmb v^*$ and $\pmb z$ to $\hat\Cal S_f$.
Then $\hatpmb v^*$ is the quadratic variation of $\hatpmb v$ (617N), the
starting value of $\hatpmb v$ is $0$ (615H)  and
$\hatpmb z=\highoverline{\exp}(\hatpmb v-\bover12\hatpmb v^*)$ (612Qb).
Also $\hatpmb v^*$ is non-decreasing, so 
$\highoverline{\exp}(\bover12\hatpmb v^*)$ is non-decreasing (614Kf).   Since
$\Cal S$ is cofinal with $\hat\Cal S_f$ (626Hc), 
$\sup_{\tau\in\hat\Cal S_f}\Expn(\highoverline{\exp}(\bover12\hat v_{\tau}))
=\sup_{\sigma\in\Cal S}\Expn(\highoverline{\exp}(\bover12 v_{\sigma}))$ is
finite.   So (i) tells us that $\hatpmb z$ is a uniformly integrable martingale
and therefore $\pmb z=\hatpmb z\restr\Cal S$ is a uniformly integrable
martingale, as claimed.   

\medskip

{\bf (b)} This is immediate from (a).
}%end of proof of 651E

\leader{651F}{Corollary} If $\pmb w$ is Brownian motion\cmmnt{ (612T)},
then $\highoverline{\exp}(\pmb w-\bover12\pmb{\iota})\restr\Cal T_b$ is
a martingale.

\proof{ As observed in 624F, $\pmb w^*$ can be identified with the
identity process $\pmb{\iota}$ on $\Cal T_f$,
and therefore lies in $L^{\infty}$ on
$\Cal T_b$, so we can apply 651Eb to $\pmb w\restr\Cal T_b\wedge\tau$ for each
$\tau\in\Cal T_b$.
}%end of proof of 651F

\leader{651G}{Theorem} Let $\Cal S$ be a non-empty
sublattice of $\Cal T$, and $\pmb v$, $\pmb w$
locally jump-free local integrators with domain $\Cal S$.   Set
$\pmb z=\highoverline{\exp}(\pmb v-v_{\downarrow}\tbf{1}-\bover12\pmb v^*)$
and $\pmb y=\pmb w-\covar{\pmb v}{\pmb w}$.   Then

\Centerline{$ii_{\pmb y\times\pmb z}(\pmb u)
=ii_{\pmb w}(\pmb u\times\pmb z)
+\iiv(\pmb u\times\pmb y\times\pmb z)$}

\noindent for any locally moderately oscillatory
process $\pmb u$ with domain $\Cal S$.

\proof{ We know that $\pmb y$ and $\pmb z$ are locally jump-free
local integrators (618T, 617I, 616N again),
so $\pmb y\times\pmb z$ also is (618Ga, 616Qa), and $\pmb u\times\pmb y$ and
$\pmb u\times\pmb z$ are locally moderately oscillatory (616Ib again, 
615F(b-iii)).  Setting $\pmb z'=\iiv(\pmb z)$,
we know that $\pmb z'-\pmb z$ is constant (651B).
If $\tau\in\Cal S$, then

$$\eqalignno{\int_{\Cal S\wedge\tau}\pmb u\,d(\pmb y\times\pmb z)
&=\int_{\Cal S\wedge\tau}\pmb u\times\pmb z\,d\pmb y
  +\int_{\Cal S\wedge\tau}\pmb u\times\pmb y\,d\pmb z
  +\int_{\Cal S\wedge\tau}\pmb u\,d\pmb yd\pmb z\cr
\displaycause{613M}
&=\int_{\Cal S\wedge\tau}\pmb u\times\pmb z\,d\pmb w
  -\int_{\Cal S\wedge\tau}\pmb u\times\pmb z\,
             d\hbox{$\covar{\pmb v}{\pmb w}$}\cr
&\mskip100mu
  +\int_{\Cal S\wedge\tau}\pmb u\times\pmb y\times\pmb z\,d\pmb v
  +\int_{\Cal S\wedge\tau}\pmb u\,d\pmb z'd\pmb y\cr
\displaycause{651B, and because the interval functions $\Delta\pmb z$ and
$\Delta\pmb z'$ are equal, so
$\Delta\pmb y\times\Delta\pmb z=\Delta\pmb z'\times\Delta\pmb y$
and  $\int_{\Cal S\wedge\tau}\ldots d\pmb yd\pmb z
=\int_{\Cal S\wedge\tau}\ldots d\pmb z'd\pmb y$}
&=\int_{\Cal S\wedge\tau}\pmb u\times\pmb z\,d\pmb w
  -\int_{\Cal S\wedge\tau}\pmb u\times\pmb z\,d\pmb vd\pmb w\cr
&\mskip100mu
  +\int_{\Cal S\wedge\tau}\pmb u\times\pmb y\times\pmb z\,d\pmb v
  +\int_{\Cal S\wedge\tau}\pmb u\times\pmb z\,d\pmb vd\pmb y\cr
\displaycause{617I once more, 617Q}
&=\int_{\Cal S\wedge\tau}\pmb u\times\pmb z\,d\pmb w
  -\int_{\Cal S\wedge\tau}\pmb u\times\pmb z\,d\pmb vd\pmb w
  +\int_{\Cal S\wedge\tau}\pmb u\times\pmb y\times\pmb z\,d\pmb v\cr
&\mskip100mu
  +\int_{\Cal S\wedge\tau}\pmb u\times\pmb z\,d\pmb wd\pmb v
  -\int_{\Cal S\wedge\tau}\pmb u\times\pmb z\,
     d\pmb vd\hbox{$\covar{\pmb w}{\pmb v}$}\cr
\displaycause{because
$\Delta\pmb v\times\Delta\pmb y=\Delta\pmb w\times\Delta\pmb v
-\Delta\covar{\pmb w}{\pmb v}\times\Delta\pmb v$}
&=\int_{\Cal S\wedge\tau}\pmb u\times\pmb z\,d\pmb w
  +\int_{\Cal S\wedge\tau}\pmb u\times\pmb y\times\pmb z\,d\pmb v
  -\int_{\Cal S\wedge\tau}\pmb u\times\pmb z\,
     d\hbox{$\covar{\pmb v}{\hbox{$\covar{\pmb w}{\pmb v}$}}$}\cr
&=\int_{\Cal S\wedge\tau}\pmb u\times\pmb z\,d\pmb w
  +\int_{\Cal S\wedge\tau}\pmb u\times\pmb y\times\pmb z\,d\pmb v\cr}$$

\noindent by 624C again, because $\pmb v$ is locally jump-free and
$\covar{\pmb w}{\pmb v}$ is locally of bounded variation, so
$\covar{\pmb v}{\covar{\pmb w}{\pmb v}}=0$.
}%end of proof of 651G

\leader{651H}{Corollary} Let $\Cal S$ be a non-empty
sublattice of $\Cal T$, and $\pmb v$, $\pmb w$
locally jump-free virtually local martingales.   Set
$\pmb z=\highoverline{\exp}(\pmb v-v_{\downarrow}\tbf{1}-\bover12\pmb v^*)$
and $\pmb y=\pmb w-\covar{\pmb w}{\pmb v}$.
Then $\pmb y\times\pmb z$ is a virtually local martingale.

\proof{ Since $\pmb v^*$ and $\covar{\pmb w}{\pmb v}$ both start from $0$,
$\pmb y\times\pmb z$ starts from a value
$w_{\downarrow}$ with finite expectation (623Kg).   By 623Kd again,
$\pmb v$ and $\pmb w$ are local integrators.   Now

\Centerline{$\pmb y\times\pmb z-w_{\downarrow}\tbf{1}
=ii_{\pmb y\times\pmb z}(\tbf 1)
=ii_{\pmb w}(\pmb z)+\iiv(\pmb y\times\pmb z)$}

\noindent is a virtually local martingale by 623O again,
so $\pmb y\times\pmb z$ also is.
}%end of proof of 651H

\leader{651I}{Proposition} Let $\Cal S$ be a non-empty sublattice of $\Cal T$
and $\pmb v=\sfamilysS{v}$, $\pmb w\cmmnt{\mskip5mu=\sfamilysS{w}}$
locally jump-free virtually local martingales such that
$\pmb z=\highoverline{\exp}(\pmb v-v_{\downarrow}\tbf{1}-\bover12\pmb v^*)$
is a uniformly integrable martingale and
$\cmmnt{v'=\mskip5mu}
\lim_{\sigma\uparrow\Cal S}(v_{\sigma}-\bover12v^*_{\sigma})$
is defined in $L^0$, where $\pmb v^*=\sfamilysS{v^*}$.   Set
$\pmb y=\pmb w-\covar{\pmb w}{\pmb v}$.
Then there is a change of law on $\frak A$ rendering $\pmb y$ a
virtually local martingale.

\proof{ Because $\pmb z$ is a uniformly integrable martingale, it
must be $\pmb{P}z\restr\Cal S$, where

\Centerline{$z
=\lim_{\sigma\uparrow\Cal S}
   \highoverline{\exp}(v_{\sigma}-v_{\downarrow}-\bover12v^*_{\sigma})
=\highoverline{\exp}(v'-v_{\downarrow})$}

\noindent
belongs to $L^1_{\bar\mu}$ (622J);  note that $\Bvalue{z>0}=1$.
Now 651H tells us that
$\pmb y\times\pmb{P}z=\pmb y\times\pmb z$ is a virtually local
$\bar\mu$-martingale.
Set $\gamma=\Expn_{\bar\mu}(z)$ and
$\bar\nu a=\bover1\gamma\Expn_{\bar\mu}(z\times\chi a)$ for $a\in\frak A$.
Then $(\frak A,\bar\nu)$ is a probability algebra and $\pmb y$ is a
virtually local $\bar\nu$-martingale, by 625C.
}%end of proof of 651I

\leader{651J}{Corollary} Let $\Cal S$ be a non-empty sublattice of $\Cal T$,
$\pmb u\cmmnt{\mskip5mu=\sfamilysS{u}}$
a locally moderately oscillatory
process such that $\gamma=\|\pmb u\|_{\infty}$ is finite,
and $\pmb w\cmmnt{\mskip5mu=\sfamilysS{w}}$ a locally jump-free
virtually local martingale with quadratic variation
$\pmb w^*$ such that $\highoverline{\exp}(\bover12\gamma^2\pmb w^*)$
is a $\|\,\|_1$-bounded process.
Set $\pmb y=\pmb w+ii_{\pmb w^*}(\pmb u)$.   Then there is a change of law
on $\frak A$ rendering $\pmb y$ a virtually local martingale.

\proof{ Set $\pmb v=-ii_{\pmb w}(\pmb u)$;  note that its starting value
is $0$.   By 618R and 623O once more, $\pmb v$ is a locally
jump-free virtually local martingale.   Express the quadratic variations
$\pmb v^*$, $\pmb w^*$ as $\family{\sigma}{\Cal S}{v^*_{\sigma}}$ and
$\family{\sigma}{\Cal S}{w^*_{\sigma}}$.   By 617Q(a-iii),

\Centerline{$v^*_{\sigma}
=\int_{\Cal S\wedge\sigma}(-\pmb u)^2\,d\pmb w^*
\le\gamma^2\int_{\Cal S\wedge\sigma}d\pmb w^*
\le\gamma^2w^*_{\sigma}$}

\noindent and
$\sup_{\sigma\in\Cal S}\Expn(\highoverline{\exp}(\bover12v^*_{\sigma}))$ is
finite.   By 651Ea,
$\pmb z=\highoverline{\exp}(\pmb v-\bover12\pmb v^*)$ is a uniformly integrable
martingale.

On the other hand,

$$\eqalignno{ii_{\pmb w^*}(\pmb u)&
=ii_{\covar{\pmb w}{\pmb w}}(\pmb u\times\tbf 1)
=\covar{ii_{\pmb w}(\pmb u)}{ii_{\pmb w}(\tbf 1)}
\Displaycause{617Q(a-ii)}
=-\covar{\pmb v}{\pmb w-w_{\downarrow}\tbf 1}
=-\covar{\pmb v}{\pmb w}\cr}$$

\noindent by 624C once more.
So $\pmb y$ is equal to $\pmb w-\covar{\pmb w}{\pmb v}$ and can be
made into a virtually local martingale by a change of law, by 651I.
}%end of proof of 651J

%\query when is $\pmb y$ a martingale?

\cmmnt{\medskip

\noindent{\bf Remark} Of course the leading application is to the case
in which $\pmb w$ is Brownian motion and $\Cal S$ is bounded above in
$\Cal T_b$, so that
$\sup_{\sigma\in\Cal S}\Expn(\highoverline{\exp}(\bover12\gamma^2w^*_{\sigma}))$
is finite for every $\gamma$.
}

\leader{651K}{\dvrocolon{S-integrals}}\cmmnt{ It is easy to find
re-formulations of 651B and 651G in terms of S-integrals, as follows.

\medskip

\noindent}{\bf Theorem} Suppose that $\family{t}{T}{\frak A_t}$ is
right-continuous, and that $\Cal S$ is a non-empty order-convex sublattice of
$\Cal T$.   Let $\pmb v=\family{\sigma}{\Cal S}{v_{\sigma}}$ be a
jump-free integrator, and $\pmb x$ an S-integrable process
with domain $\Cal S$.   Set
$\pmb z=\highoverline{\exp}(\pmb v-v_{\downarrow}\tbf{1}-\bover12\pmb v^*)$.

(a)

\Centerline{$\Sint_{\Cal S}\pmb x\,d\pmb z
=\Sint_{\Cal S}\pmb x\times\pmb z\,d\pmb v$.}

(b) Suppose that $\pmb w$ is another jump-free integrator with domain
$\Cal S$.   Set $\pmb y=\pmb w-\covar{\pmb v}{\pmb w}$.   Then

\Centerline{$\Sint_{\Cal S}\pmb x\,d(\pmb y\times\pmb z)
=\Sint_{\Cal S}\pmb x\times\pmb z\,d\pmb w
+\Sint_{\Cal S}\pmb x\times\pmb y\times\pmb z\,d\pmb v$.}

\proof{{\bf (a)} We know that $\pmb z$ is jump-free
(651B).   So $\pmb z\times\tbfoneminus{\Cal S}=\pmb z_<$ (641O)
and $\pmb z$ and $\pmb x\times\pmb z$ are S-integrable (645F, 645Ka, 645Pb).   
Next,

$$\eqalignno{\Siiv(\pmb z)
&=\Siiv(\pmb z\times\tbfoneminus{\Cal S})
\Displaycause{646Kb}
=\Siiv(\pmb z_<)
=\iiv(\pmb z)
\Displaycause{646Kc}
=\pmb z-\tbf 1
\cr}$$

\noindent (651B).   So 646R tells us that

\Centerline{$\Sint_{\Cal S}\pmb x\times\pmb z\,d\pmb v
=\Sint_{\Cal S}\pmb x\,d(\Siiv(\pmb z))
=\Sint_{\Cal S}\pmb x\,d(\pmb z-\tbf 1)
=\Sint_{\Cal S}\pmb x\,d\pmb z-\Sint_{\Cal S}\pmb x\,d\tbf 1
=\Sint_{\Cal S}\pmb x\,d\pmb z$.}

\medskip

{\bf (b)} Similarly,

$$\eqalignno{&\Sint_{\Cal S}\pmb x\times\pmb z\,d\pmb w
   +\Sint_{\Cal S}\pmb x\times\pmb y\times\pmb z\,d\pmb v
=\Sint_{\Cal S}\pmb x\,d(\Sii_{\pmb w}(\pmb z))
   +\Sint_{\Cal S}\pmb x\,d(\Siiv(\pmb y\times\pmb z))
\Displaycause{because $\pmb y$ and $\pmb y\times\pmb z$ are jump-free}
\mskip50mu=\Sint_{\Cal S}\pmb x\,d(ii_{\pmb w}(\pmb z))
   +\Sint_{\Cal S}\pmb x\,d(\iiv(\pmb y\times\pmb z))
\Displaycause{again because $\pmb y\times\pmb z$ is jump-free}
\mskip50mu
 =\Sint_{\Cal S}\pmb x\,d(ii_{\pmb w}(\pmb z)+\iiv(\pmb y\times\pmb z)).
\cr}$$

\noindent But setting $\pmb u=\tbf 1^{(\Cal S)}$ in 651G, we have

\Centerline{$ii_{\pmb w}(\pmb z)+\iiv(\pmb y\times\pmb z)
=ii_{\pmb y\times\pmb z}(\tbf 1^{(\Cal S)})
=\pmb y\times\pmb z-w_{\downarrow}\tbf 1^{(\Cal S)}$}

\noindent where $w_{\downarrow}$ is the common starting value of $\pmb w$, 
$\pmb y$ and $\pmb y\times\pmb z$.   So

$$\eqalign{\Sint_{\Cal S}\pmb x\times\pmb z\,d\pmb w
   +\Sint_{\Cal S}\pmb x\times\pmb y\times\pmb z\,d\pmb v
&=\Sint_{\Cal S}\pmb x\,d(\pmb y\times\pmb z)
   -\Sint_{\Cal S}\pmb x\,d(w_{\downarrow}\tbf 1^{(\Cal S)})
\cr&=\Sint_{\Cal S}\pmb x\,d(\pmb y\times\pmb z),
\cr}$$

\noindent as claimed.
}%end of proof of 651K

\exercises{\leader{651X}{Basic exercises (a)}
%\spheader 651Xa
Let $\Cal S$ be a non-empty sublattice of $\Cal T$.   Suppose that
$\pmb v=\family{\sigma}{\Cal S}{v_{\sigma}}$ is a locally
jump-free local integrator, and
$y\in\bigcap_{\sigma\in\Cal S}L^0(\frak A_{\sigma})$.
Set $v_{\downarrow}=\lim_{\sigma\downarrow\Cal S}v_{\sigma}$ and
$\pmb z
=y\,\highoverline{\exp}(\pmb v-v_{\downarrow}\tbf{1}-\bover12\pmb v^*)$.
Show that $\pmb z=y\tbf{1}+\iiv(\pmb z)$.
%651B

%\leader{651Y}{Further exercises (a)}

}%end of exercises

\endnotes{
\Notesheader{651} We are
trying to combine the fundamental result 623O, which tells us we have a
virtually local martingale, with 625B,
which will demand a true martingale if we are
to identify our exponential process
$\pmb z=\highoverline{\exp}(\pmb v-v_{\downarrow}\tbf{1}-\bover12\pmb v^*)$
with $\pmb{P}z$ for
a Radon-Nikod\'ym derivative $z$.   So a good deal of work has been done on
this, of which I offer a sample in 651D-651E.   In this context Brownian
motion (651F) is more than the leading example;  it is the archetype of
a locally jump-free martingale, in a sense which I hope
to make clear in \S653.

Note that $\pmb z$ can also be regarded as a solution of the stochastic
integral equation

\Centerline{$z_{\tau}=\chi 1+\int_{\Cal S\wedge\tau}\pmb z\,d\pmb v$}

\noindent (651B).
The correction term $-\bover12\pmb v^*$ in the formula for
$\pmb z$ vanishes if $\pmb v$ is of bounded variation, leaving us with
the solution $z_{\tau}=\highoverline{\exp}(v_{\tau})$
corresponding to the solution of the integral equation

\Centerline{$z(t)=1+\int_0^tz(s)v'(s)ds$}

\noindent when $v(0)=0$.   For martingales $\pmb v$ we need the term in
$\pmb v^*$ here just as we need it in It\^o's Lemma.   I will return to
integral equations in \S654.
}%end of notes

\discrpage

\frfilename{mt652.tex}
\versiondate{7.1.23/18.2.23}
\copyrightdate{2018}

\def\chaptername{Applications}
\def\sectionname{L\'evy processes}

\newsection{652} When defining the Poisson process in 612U,
I referred to 455P in Volume 4.   \S455 is hard work;  it is the longest section
in the whole treatise and bristles with technical difficulties.
Some of these are
exacerbated by the generality which seemed natural at that point
-- for instance,
it is meant to support the treatment of multidimensional Brownian motion in
Chapter 47.   The processes described in the last quarter of
\S455 take values in Polish groups which
need not even be abelian.   But these processes have always been the
standard-bearers for the theory of stochastic processes I have set out to
describe in the present volume, and the time has come to link the approaches.

\cmmnt{\leader{652A}{Notation} If \Asissix\
is a stochastic integration structure,
I write $\check t$ for the constant stopping time at $t$, for each
$t\in T$, $\check T$ for the sublattice $\{\check t:t\in T\}$ of $\Cal T$, and
$\Cal T_b$, $\Cal T_f\subseteq\Cal T$ for the ideals of bounded and
finite-valued stopping times.
$\mu_L$ will be Lebesgue measure on $\Bbb R$.   For any measure $\mu$,
$\Cal N(\mu)$ will be the ideal of $\mu$-negligible sets.   For a topological
space $\Omega$, $\Cal B(\Omega)$ will be the Borel $\sigma$-algebra of $\Omega$.
If $\Cal S$ is a sublattice of $\Cal T$, $\Cal I(\Cal S)$ will be the set of
finite sublattices of $\Cal S$, and $\Cal S^{2\uparrow}$ will be
$\{(\sigma,\tau):\sigma$, $\tau\in\Cal S$, $\sigma\le\tau\}$.
}

\leader{652B}{Independence (a)}\cmmnt{ Since Chapter 27
I have given the fundamental concept of
`stochastic independence' rather cavalier treatment.   So perhaps I should run
through the definitions in the forms I will use here.}   If $(\frak A,\bar\mu)$
is a probability algebra, two subalgebras $\frak B$, $\frak C$ of $\frak A$
are (stochastically) independent
if $\bar\mu(b\Bcap c)=\bar\mu b\cdot\bar\mu c$ whenever $b\in\frak B$ and
$c\in\frak C$;  more generally, a family $\familyiI{\frak B_i}$ of subalgebras
of $\frak A$ is independent if
$\bar\mu(\inf_{i\in J}b_i)=\prod_{i\in J}\bar\mu b_i$ whenever $J\subseteq I$ is
finite and $b_i\in\frak B_i$ for every $i\in J$\cmmnt{ (325L, 458La)}.
Turning to families in $L^0=L^0(\frak A)$, $\familyiI{u_i}$ is independent if
$\familyiI{\frak B_i}$ is independent where
$\frak B_i=\{\Bvalue{u_i\in E}:E\subseteq\Bbb R$ is Borel$\}$
is the closed subalgebra generated by $u_i$ for each $i\in I$.
$u\in L^0$ is independent of an algebra $\frak C\subseteq\frak A$ if
$\frak B$ and $\frak C$ are independent where $\frak B$ is the closed subalgebra
generated by $u$.   More elaborately, a family $\familyiI{u_i}$ in $L^0$ is
independent of $\frak C$, where $\frak C$ is a subalgebra of $\frak A$, if
$\frak B$ and $\frak C$ are independent, where $\frak B$ is the smallest
closed subalgebra of $\frak A$ including all the subalgebras generated by the
$u_i$.

\medskip

{\bf (b)} Using the Monotone Class Theorem\cmmnt{ in the form 313Gc},
we see that
if $B$, $C\subseteq\frak A$ are such that both $B$ and $C$ are closed under
$\Bcap$ and
$\bar\mu(b\Bcap c)=\bar\mu b\cdot\bar\mu c$ for all $b\in B$ and $c\in C$,
then the closed subalgebras $\frak B$, $\frak C$ generated by $B$, $C$
respectively are independent.   \prooflet{(Show first that
$\bar\mu(b\Bcap c)=\bar\mu b\cdot\bar\mu c$ for $b\in B$ and $c\in\frak C$.)}

\medskip

{\bf (c)} If $\frak B$ is a closed subalgebra of $\frak A$, the set
$C=\{u:u\in L^0$, $u$ is independent of $\frak B\}$ is closed for the
topology of convergence in measure.   \prooflet{\Prf\ Take $v\in\overline{C}$.
$\alpha\in\Bbb R$ and $\epsilon>0$.   Write $d$ for $\Bvalue{v>\alpha}$.
Then there are a $\delta>0$ such that
$\bar\mu(d\Bsetminus\Bvalue{v>\alpha+2\delta})\le\epsilon$, and
a $u\in C$ such that $\theta(v-u)\le\epsilon\delta$.
Consider $c=\Bvalue{u>\alpha+\delta}$.
Then $c\Bsetminus d\Bsubseteq\Bvalue{u-v>\delta}$ and

$$\eqalign{d\Bsetminus c&
\Bsubseteq(d\Bsetminus\Bvalue{v>\alpha+2\delta})
   \Bcup(\Bvalue{v>\alpha+2\delta}\Bsetminus c)\cr&
\Bsubseteq(d\Bsetminus\Bvalue{v>\alpha+2\delta})
   \Bcup\Bvalue{v-u>\delta},\cr}$$

\noindent so

\Centerline{$\bar\mu(c\Bsymmdiff d)
\le\bar\mu(d\Bsetminus\Bvalue{v>\alpha+2\delta})
   +\bar\mu\Bvalue{|u-v|>\delta}
\le 2\epsilon$.}

Now if $b\in\frak B$,

$$\eqalignno{|\bar\mu(b\Bcap d)-\bar\mu b\cdot\bar\mu d|&
\le|\bar\mu(b\Bcap d)-\bar\mu(b\Bcap c)|
  +\bar\mu b|\bar\mu d-\bar\mu c|
  +|\bar\mu(b\Bcap c)
       -\bar\mu b\cdot\bar\mu c|\cr&
\le 2\bar\mu(d\symmdiff c)+0
\Displaycause{because $u$ is independent of $\frak B$}
\le 4\epsilon.\cr}$$

\noindent As $\epsilon>0$, $\bar\mu(b\Bcap d)=\bar\mu b\cdot\bar\mu d$;
as $\alpha$ and $b$ are arbitrary, $u$ is independent of $\frak B$;  as $u$ is
arbitrary, $C$ is closed.\ \Qed
}

\leader{652C}{Definition} Let \Asissixt\ be a
right-continuous real-time stochastic integration structure.   I will say that a
fully adapted process $\family{\sigma}{\Cal T_f}{v_{\sigma}}$ is a
{\bf L\'evy process} if it is locally near-simple and

\inset{\noindent whenever $s$, $t\ge 0$,
$v_{(s+t)\varCspcheck}-v_{\check s}$ is independent of $\frak A_s$
and has the same distribution\cmmnt{ (364Gb\formerly{3{}64Xd})} as
$v_{\check t}$.}

\medskip

\noindent{\bf Examples (i)} The identity process is a L\'evy process.
\prooflet{(By 613Ea, it is locally near-simple.   And
in the notation of 612F, $\iota_{\check t}=t\chi 1$ for every $t\ge 0$.)}

\medskip

{\bf (ii)} Brownian motion\cmmnt{, as described in 612T},
is a L\'evy process\prooflet{ (477D(c-ii))}.

\medskip

{\bf (iii)} The standard Poisson process\cmmnt{, as described in 612U,}
is a L\'evy process\prooflet{ (455P-455R)}.  %455P 455Q 455R

\cmmnt{\medskip

{\bf (iv)} The Cauchy process, to be described in 652M-652O below, is a
L\'evy process.
}

\leader{652D}{Lemma} If \Asissixt\ is a right-continuous real-time
stochastic integration structure and
$\pmb v=\family{\sigma}{\Cal T_f}{v_{\sigma}}$ a L\'evy process, then
$v_{\check 0}=\lim_{t\downarrow 0}v_{\check t}=0$.

\proof{ $v_{\check 0}$ has the same distribution as
$v_{\check 0}-v_{\check 0}$, so must be $0$.   And $\check 0=\inf_{t>0}\check t$
in $\Cal T$ (611Ce), so
$\lim_{t\downarrow\ocint{0,\infty}}v_{\check t}=v_{\check 0}$ by 632E.
}%end of proof of 652D

\leader{652E}{Proposition} Let \Asissixt\ be a
right-continuous real-time stochastic integration structure, and
$\pmb v=\family{\tau}{\Cal T_f}{v_{\tau}}$,
$\pmb w=\family{\tau}{\Cal T_f}{w_{\tau}}$ two L\'evy processes.   If
$v_{\check t}=w_{\check t}$ for every $t\ge 0$, then $\pmb v=\pmb w$.

\proof{ Note that $\pmb v\restr\Cal T_b$ is locally near-simple, just because
$\pmb v\restr\Cal T_b\wedge\tau=\pmb v\restr\Cal T_f\wedge\tau$ is near-simple
for every $\tau\in\Cal T_b$.   Similarly, $\pmb w\restr\Cal T_b$ is near-simple.
Now $\{\check t:t\ge 0\}$ separates $\Cal T_b$ (633Da)
and is cofinal with $\Cal T_b$, so $\pmb v\restr\Cal T_b=\pmb w\restr\Cal T_b$
by 633F.   As $\Cal T_f$ is the covered envelope of $\Cal T_b$,
$\pmb v=\pmb w$ (612Qa).
}%end of proof of 652E

\leader{652F}{Classical L\'evy \dvrocolon{processes}}\cmmnt{ I have already
used the phrase `L\'evy process' in \S455 to describe a class of processes
of the form
$\langle X_t\rangle_{t\ge 0}$ (455Q).   In one sense these are very much more
general than the processes of this section, because they deal with $U$-valued
random variables where $U$ can be any separable metrizable topological group.
But there is a crucial difference in that the whole analysis is based on
a prior notion of what the distributions $\lambda_t$ of the $X_t$ are to be.
In particular, both the measure $\ddot\mu$ and the
filtration $\langle\hat{\ddot\Sigma}_t\rangle_{t\ge 0}$ (455T)
are defined from the joint distribution of $\langle X_t\rangle_{t\ge 0}$ (454K),
while in the present section I am dealing with processes in
which the underlying probability algebra $(\frak A,\bar\mu)$
and the filtration $\langle\frak A_t\rangle_{t\ge 0}$ need not be describable
from the process $\langle v_{\check t}\rangle_{t\ge 0}$.

To bridge the gap, I offer the following.

\medskip

\noindent }{\bf Proposition}
Let $\Cdlg$ be the space of \cadlag\ real-valued functions
on $\coint{0,\infty}$, endowed with its topology of pointwise convergence.
Let $\langle\lambda_t\rangle_{t>0}$ be a family
of distributions\cmmnt{ (that is,
Radon probability measures on $\Bbb R$)} such that the convolution
$\lambda_s*\lambda_t$\cmmnt{ (257A)} is equal to $\lambda_{s+t}$ for all
$s$, $t>0$, and $\lim_{t\downarrow 0}\lambda_tG=1$ for every open subset $G$
of $\Bbb R$ containing $0$.   Let $\ddot\mu$ be the completion regular
quasi-Radon probability measure on $\Cdlg$ defined by saying that

$$\eqalignno{\ddot\mu\{\omega:\omega\in\Cdlg,\,
\omega(s_0)\in E_0&,\,\omega(s_i)-\omega(s_{i-1})\in E_i
\text{ for }1\le i\le n\}\cr&
=\delta_0E_0\cdot\prod_{i=1}^n\lambda_{s_i-s_{i-1}}E_i&(*)\cr}$$

\noindent whenever $0=s_0<\ldots<s_n$ in $\coint{0,\infty}$ and
$E_0,\ldots,E_n\subseteq\Bbb R$ are Borel sets\cmmnt{ (here
$\delta_0$ is the Dirac measure concentrated at $0$)}, and $\ddot\Sigma$ its
domain.   For $t\ge 0$, set

\Centerline{$\ddot\Sigma_t
=\{F:F\in\ddot\Sigma$, $\omega'\in F$ whenever $\omega\in F$,
$\omega'\in\Cdlg$ and $\omega'\restr[0,t]=\omega\restr[0,t]\}$,}

\Centerline{$\hat{\ddot\Sigma}_t=\{F\symmdiff A:F\in\ddot\Sigma_t$,
$A\in\Cal N(\ddot\mu)\}$}

\noindent so that $\langle\hat{\ddot\Sigma}_t\rangle_{t\ge 0}$
is a right-continuous
filtration of $\sigma$-subalgebras of $\ddot\Sigma$\cmmnt{ (455T)}.   Let
$(\frak C,\bar{\ddot\mu})$ be the measure algebra of
$(\Cdlg,\ddot\Sigma,\ddot\mu)$ and
set $\frak C_t=\{E^{\ssbullet}:E\in\hat{\ddot\Sigma}_t\}$  for $t\ge 0$, so that
$(\frak C,\bar{\ddot\mu},\coint{0,\infty},\langle\frak C_t\rangle_{t\ge 0})$
is a right-continuous stochastic integration
structure\prooflet{ (632K)} with associated set $\Cal T$ of stopping times
and family $\family{\tau}{\Cal T}{\frak C_{\tau}}$ of closed subalgebras.
Setting $X_t(\omega)=\omega(t)$ for $\omega\in\Omega$ and $t\ge 0$,
$\langle X_t\rangle_{t\ge 0}$ is progressively measurable and gives rise to
a locally near-simple process
$\pmb u=\family{\sigma}{\Cal T_f}{u_{\sigma}}$ such that
if $\sigma\in\Cal T_f$ is represented by
a stopping time $h:\Cdlg\to\coint{0,\infty}$\cmmnt{ (in
the sense of 455L or 612Gb)} adapted to
$\langle\hat{\ddot\Sigma}_t\rangle_{t\ge 0}$, and
$X_h(\omega)=X_{h(\omega)}(\omega)$ for $\omega\in\Cdlg$, then
$u_{\sigma}=X_h^{\ssbullet}$ in $L^0(\mu)$\cmmnt{ (612H, 631D)}.
Now $\pmb u$ is a L\'evy process\cmmnt{ as defined in 652C},
and the distribution of $u_{\check t}$ is $\lambda_t$ for every $t\ge 0$.

\proof{{\bf (a)} For the existence of such a measure $\ddot\mu$, see 455Pc and
455K.   I did not discuss the question of uniqueness there, but
as $\ddot\mu$ is complete and completion regular it must be the completion of
its restriction $\ddot\mu\restr\CalBa(\Cdlg)$
to the Baire $\sigma$-algebra of $\Cdlg$.   Now $\CalBa(\Cdlg)$ is the subspace
$\sigma$-algebra on $\Cdlg$ induced by
$\CalBa(\BbbR^{\coint{0,\infty}}=\Tensorhat_{\coint{0,\infty}}\Cal B(\Bbb R)$
(4A3N), so $\CalBa(\Cdlg)$ is the $\sigma$-algebra of subsets of $\Cdlg$
generated by the coordinate functionals $\omega\mapsto\omega(s)$ for
$s\ge 0$, and by the Monotone Class Theorem (136C)
$\ddot\mu\restr\CalBa(\Cdlg)$ is
determined by the values specified in (*) above.

\medskip

{\bf (b)} Theorem 455U tells us that if $h:\Cdlg\to[0,\infty]$
is a stopping time
adapted to $\langle\hat{\ddot\Sigma}_t\rangle_{t\ge 0}$ and we define
$\phi_h:\Cdlg\times\Cdlg\to\Cdlg$ by setting

$$\eqalign{\phi_h(\omega,\omega')(t)
&=\omega'(t-h(\omega))+\omega(h(\omega))
   \text{ if }t\ge h(\omega),\cr
&=\omega(t)\text{ otherwise},\cr}$$

\noindent then $\phi_h$ is \imp\ for $\ddot\mu\times\ddot\mu$ and $\ddot\mu$.

\medskip

{\bf (c)} Turning to the conditions in 652C, set $X_t(\omega)=\omega(t)$ for
$\omega\in\Cdlg$ and $t\ge 0$.   For $t\ge 0$, $X_t$ is continuous, so is
$\ddot\Sigma$-measurable, and now $X_t$ is $\hat{\ddot\Sigma}_t$-measurable.
By 631D, $\langle X_t\rangle_{t\ge 0}$ is progressively measurable so $\pmb u$
is well-defined and locally near-simple (631D).

If $s$, $t\ge 0$ then
the distribution of $u_{(s+t)\varCspcheck}-u_{\check s}$ is the distribution of
the random variable $\omega\mapsto\omega(s+t)-\omega(s)$.   But now take
$h$ to be the constant stopping time with value $s$ and consider
$\phi_h:\Cdlg\times\Cdlg\to\Cdlg$.   We have

\Centerline{$\phi_h(\omega,\omega')(s+t)-\phi_h(\omega,\omega')(s)
=\phi_h(\omega,\omega')(s+t)-\omega(s)
=\omega'(t)$.}

\noindent Because $\phi_h$ is \imp, the distribution of
$\omega\mapsto\omega(s+t)-\omega(s)$ is the same as the distribution of
$(\omega,\omega')\mapsto
\phi_h(\omega,\omega')(s+t)-\phi_h(\omega,\omega')(s)=\omega'(t)$, which is the
same as the distribution of $\omega'\mapsto\omega'(t)$ and $u_{\check t}$.

Finally, if $s\le t$, $E\in\hat{\ddot\Sigma}_s$ and $F\subseteq\Bbb R$ is Borel,
then take $h$ once again to be the constant stopping time with value $s$, and
let $E'\in\ddot\Sigma_t$ be such that $\ddot\mu(E\symmdiff E')=0$.   Then

$$\eqalign{\ddot\mu(E\cap\{&\omega:\omega(t)-\omega(s)\in F\}\cr&
=\ddot\mu(E'\cap\{\omega:\omega(t)-\omega(s)\in F\}\cr&
=\ddot\mu^2\{(\omega,\omega'):\phi_h(\omega,\omega')\in E',\,
   \phi_h(\omega,\omega')(t)-\phi_h(\omega,\omega')(s)\in F\}\cr&
=\ddot\mu\{\omega:\omega\in E'\}
   \cdot\ddot\mu\{\omega':\omega'(t-s)\in F\}\cr&
=\ddot\mu\{\omega:\omega\in E\}
   \cdot\ddot\mu\{\omega':\omega'(t)-\omega'(s)\in F\}.\cr}$$

\noindent Translating this into terms of
$(\frak C,\bar{\ddot\mu},\coint{0,\infty},\langle\frak C_t\rangle_{t\ge 0})$
and $\pmb u$,

\Centerline{$\bar{\ddot\mu}(a\Bcap\Bvalue{u_{\check t}-u_{\check s}\in F})
=\bar{\ddot\mu}a\cdot\bar{\ddot\mu}\Bvalue{u_{\check t}-u_{\check s}\in F}$}

\noindent for every $a\in\frak C_s$ and Borel set $F\subseteq\Bbb R$;  that is,
$u_{\check t}-u_{\check s}$ is independent of $\frak C_s$.   So $\pmb u$
satisfies all the clauses of 652C.
}%end of proof of 652F

\leader{652G}{Sums of stopping times} Let \Asissixt\ be a real-time stochastic
integration structure.

(a) If $\tau\in\Cal T$ and $s\ge 0$ we have an element $\tau+\check s$ of
$\Cal T$ defined by saying that

$$\eqalign{\Bvalue{\tau+\check s>t}
&=\Bvalue{\tau>t-s}\text{ if }s\le t,\cr
&=1\text{ if }s>t.\cr}$$

(b)(i) $\tau\vee\check s\le\tau+\check s$ whenever $\tau\in\Cal T$ and
$s\ge 0$.

\quad(ii) If $\tau\in\Cal T$ and $s$, $s'\ge 0$ then
$\tau+(s+s')\varCspcheck=(\tau+\check s)+\check s'$.

\quad(iii) $\tau+\check 0=\tau$ for every $\tau\in\Cal T$.

\quad(iv) If $s$, $s'\ge 0$ then $\check s+\check s'=(s+s')\varCspcheck$.

\quad(v) If $\tau$, $\tau'\in\Cal T$ and $s\ge 0$ then
$\Bvalue{\tau\le\tau'}\Bsubseteq\Bvalue{\tau+\check s\le\tau'+\check s}$.

\quad(vi) For any $\tau\in\Cal T$, $\{\tau+\check s:s\ge 0\}$
separates $\Cal T\vee\tau$.

\quad(vii) Suppose that $\langle\frak A_t\rangle_{t\ge 0}$ is right-continuous.
If $A\subseteq\Cal T$, $s\ge 0$ and we write $A+\check s$ for
$\{\sigma+\check s:\sigma\in A\}$, then $\inf(A+\check s)=(\inf A)+\check s$.

\proof{{\bf (a)} Setting

\Centerline{$a_t=\Bvalue{\tau>t-s}$ if $s\le t$,\quad $1$ otherwise,}

\noindent we see that $a_t$ belongs to $\frak A_{t-s}\subseteq\frak A_t$ in the
first case and to $\frak A_t$ in either case.   If $0\le t\le t'$, then

$$\eqalign{a_t
&=\Bvalue{\tau>t-s}\Bsupseteq\Bvalue{\tau>t'-s}=a_{t'}\text{ if }s\le t,\cr
&=1\Bsupseteq\Bvalue{\tau>t'-s}=a_{t'}\text{ if }t<s\le t',\cr
&=1=a_{t'}\text{ if }t'<s,\cr}$$

\noindent so $a_t\Bsupseteq a_{t'}$ in all cases.   And

$$\eqalign{\sup_{t'>t}a_{t'}=\lim_{t'\downarrow t}a_{t'}
&=\lim_{t'\downarrow t}\Bvalue{\tau>t'-s}=\Bvalue{\tau>t-s}=a_t
   \text{ if }s\le t,\cr
&=\lim_{t'\downarrow t,t'<s}a_{t'}=1=a_t\text{ otherwise}.\cr}$$

Thus $\langle a_t\rangle_{t\ge 0}$ satisfies the conditions of 611A(b-i) and
defines a stopping time which we may call $\tau+\check t$.

\medskip

{\bf (b)(i)} For any $t\ge 0$,

$$\eqalign{\Bvalue{\tau\vee\check s>t}
=\Bvalue{\tau>t}\Bcup\Bvalue{\check s>t}
&\Bsubseteq\Bvalue{\tau>t-s}=\Bvalue{\tau+\check s>t}\text{ if }s\le t,\cr
&\Bsubseteq 1=\Bvalue{\tau+\check s>t}\text{ if }s\ge t.\cr}$$

\medskip

\quad{\bf (ii)}

$$\eqalign{\Bvalue{(\tau+\check s)+\check s'>t}&
=\Bvalue{\tau+\check s>t-s'}=\Bvalue{\tau>t-(s+s')}\cr&
\mskip 100mu
   =\Bvalue{\tau+(s+s')\varCspcheck}\text { if }s+s'\le t,\cr&
=1=\Bvalue{\tau+(s+s')\varCspcheck}\cr&
\mskip100mu
   \text{ if }t<s\text{ or }s\le t\text{ and }t-s<s'.\cr}$$

\medskip

\quad {\bf (iii)} The formula in (a) tells us that
$\Bvalue{\tau+\check 0>t}=\Bvalue{\tau>t-0}=\Bvalue{\tau>t}$ for every $t$.

\medskip

\quad{\bf (iv)} For $t\ge 0$,

$$\eqalign{\Bvalue{\check s+\check s'>t}
&=\Bvalue{\check s>t-s'}=0=\Bvalue{(s+s')\varCspcheck>t}
     \text{ if }s+s'\le t,\cr
&\mskip100mu\text{i.e., }s'\le t\text{ and }s\le t-s',\cr
&=\Bvalue{\check s>t-s'}=1=\Bvalue{(s+s')\varCspcheck>t}
     \text{ if }s'\le t<s+s',\cr
&\mskip100mu\text{i.e., }s'\le t\text{ and }t-s'<s,\cr
&=1=\Bvalue{(s+s')\varCspcheck>t}\text{ if }t<s'.\cr}$$

\medskip

\quad{\bf (v)} If $0\le t<s$ then
$\Bvalue{\tau'+\check s>t}=1$.   If $s\le t$ then

\Centerline{$\Bvalue{\tau\le\tau'}\Bcap\Bvalue{\tau+\check s>t}
=\Bvalue{\tau\le\tau'}\Bcap\Bvalue{\tau>t-s}
\Bsubseteq\Bvalue{\tau'>t-s}=\Bvalue{\tau'+\check s>t}$.}

\noindent So $\Bvalue{\tau+\check s>t}\Bsetminus\Bvalue{\tau'+\check s>t}$ does
not meet $\Bvalue{\tau\le\tau'}$ for any $t>0$,
$\Bvalue{\tau\le\tau'}\Bcap\Bvalue{\tau'+\check s<\tau+\check s}=0$
(see 611D) and
$\Bvalue{\tau\le\tau'}\Bsubseteq\Bvalue{\tau+\check s\le\tau'+\check s}$.

\medskip

\quad{\bf (vi)} Suppose that $\tau'$, $\tau''\in\Cal T\vee\tau$ and
$0\ne a\Bsubseteq\Bvalue{\tau'<\tau''}$.   Then there are a $t\ge 0$ such that
$b=a\Bcap\Bvalue{\tau''>t}\Bsetminus\Bvalue{\tau'>t}$ is non-zero, and a
$t'>t$ such that $c=b\Bsetminus\Bvalue{\tau'>t}$ is non-zero.
For $i\in\Bbb N$ set $s_i=i(t'-t)$.   Then there is certainly an $i$
such that
$i(t'-t)>t$, in which case $\Bvalue{\tau+\check s_i>t}=1$, so there is a first
$n\in\Bbb N$ such that $d=c\Bcap\Bvalue{\check t<\tau+\check s_n}$ is
non-zero.
Since $\Bvalue{\tau+\check s_0>t}\Bsubseteq\Bvalue{\tau'>t}$ is disjoint from
$c$, $n\ge 1$.

Now $c\Bsubseteq\Bvalue{\tau'\le\check t}$ and
$d\Bsubseteq\Bvalue{\check t<\tau+\check s_n}$, so
$d\Bsubseteq\Bvalue{\tau'\le\tau+\check s_n}$.   At the same time,

\Centerline{$c\Bsubseteq\Bvalue{\tau+\check s_{n-1}\le\check t}
\Bsubseteq\Bvalue{\tau+\check s_n\le\check t'}$}

\noindent by (ii) and (v) above,
and as $c$ is also included in $\Bvalue{\check t'<\tau''}$,
we have $c\Bsubseteq\Bvalue{\tau+\check s_n<\tau''}$ and

\Centerline{$a\Bcap\Bvalue{\tau'\le\tau+\check s_n}
   \Bcap\Bvalue{\tau+\check s_n<\tau''}
\Bsupseteq d\ne 0$.}

\noindent As $a$ is arbitrary,
$\Bvalue{\tau'<\tau''}\Bsubseteq\sup_{s\ge 0}
  (\Bvalue{\tau'\le\tau+\check s}\Bcap\Bvalue{\tau+\check s<\tau''})$.
As $\tau'$ and $\tau''$ are arbitrary, we have the result.

\medskip

\quad{\bf (vii)} Write $\tau$ for $\inf A$, $\tau'$ for $\inf(A+\check s)$.
If $A$ is empty, so is $A+\check s$, so $\max\Cal T=\tau=\tau'=\tau+\check s$.
Otherwise. if $0\le t<s$ then

\Centerline{$\Bvalue{\tau+\check s>t}=1=\Bvalue{\check s>t}
=\Bvalue{\tau'>t}$}

\noindent because $\check s$ is a lower bound for $A+\check s$ so
$\check s\le\tau'$.   If $A\ne\emptyset$ and $s\le t$ then

$$\eqalignno{\Bvalue{\tau'>t}
&=\sup_{t'>t}\inf_{\sigma\in A+\check s}\Bvalue{\sigma>t'}
\Displaycause{632C(a-i)}
=\sup_{t'>t}\inf_{\sigma\in A}\Bvalue{\sigma+\check s>t'}
=\sup_{t'>t}\inf_{\sigma\in A}\Bvalue{\sigma>t'-s}\cr&
=\sup_{t'>t-s}\inf_{\sigma\in A}\Bvalue{\sigma>t'}
=\Bvalue{\tau>t-s}
\Displaycause{632C(a-i) again}
=\Bvalue{\tau+\check s>t}.\cr}$$

\noindent Thus $\Bvalue{\tau'>t}=\Bvalue{\tau+\check s>t}$ for all $t\ge 0$ and
$\tau'=\tau+\check s$, as claimed.
}%end of proof of 652G

\leader{652H}{Proposition} Let \Asissixt\ be a right-continuous real-time
stochastic
integration structure, and $\pmb v=\family{\sigma}{\Cal T_f}{v_{\sigma}}$
a L\'evy process.

(a) For any $s\ge 0$ and $\tau\in\Cal T_f$,
$v_{\tau+\check s}-v_{\tau}$ is independent of $\frak A_{\tau}$ and has the same
distribution as $v_{\check s}$.

(b) For any $\tau\in\Cal T_f$,
$\langle v_{\tau+\check s}-v_{\tau}\rangle_{s\ge 0}$ is independent of
$\frak A_{\tau}$ and has the same
distribution as $\langle v_{\check s}\rangle_{s\ge 0}$.

\proof{{\bf (a)(i)} Write $D$ for the set of those $\tau\in\Cal T_f$ such that
$v_{\tau+\check s}-v_{\tau}$ is independent of $\frak A_{\tau}$ and has the same
distribution as $v_{\check s}$.   By the definition in 652C,
$\check T$ is included in $D$.

\medskip

\quad{\bf (ii)} $D$ is full.   \Prf\ Suppose that $\tau\in\Cal T$ and
$\sup_{\sigma\in D}\Bvalue{\tau=\sigma}=1$.   Then $\tau\in\Cal T_f$ by
611N(e-i), and there is a sequence $\sequence{i}{\sigma_i}$ in $D$ such that
$1=\sup_{i\in\Bbb N}a_i$, where $a_i=\Bvalue{\tau=\sigma_i}$ for $i\in\Bbb N$.
Set $b_i=a_i\Bsetminus\sup_{j<i}a_j$ for $i\in\Bbb N$;  as
$a_i\in\frak A_{\tau}\cap\frak A_{\sigma_i}$ for every $i$,
$b_i\in\frak A_{\tau}$ and therefore $b_i\in\frak A_{\sigma_i}$ (611H(c-iii)).

Now if $a\in\frak A_{\tau}$ and $\alpha\in\Bbb R$,

$$\eqalignno{\bar\mu(a\Bcap\Bvalue{v_{\tau+\check s}-v_{\tau}>\alpha})&
=\sum_{i=0}^{\infty}\bar\mu(a\Bcap b_i
   \Bcap\Bvalue{v_{\tau+\check s}-v_{\tau}>\alpha})\cr&
=\sum_{i=0}^{\infty}\bar\mu(a\Bcap b_i
   \Bcap\Bvalue{v_{\sigma_i+\check s}-v_{\sigma_i}>\alpha})
\Displaycause{because $b_i\Bsubseteq\Bvalue{v_{\tau}=v_{\check s}}
\Bcap\Bvalue{\tau+\check s=\sigma_i+\check s}$, by 652G(b-v)}
=\sum_{i=0}^{\infty}\bar\mu(a\Bcap b_i)\cdot
   \bar\mu\Bvalue{v_{\sigma_i+\check s}-v_{\sigma_i}>\alpha}
\Displaycause{because $a\Bcap b_i\in\frak A_{\sigma_i}$ and $\sigma_i\in D$, so
$v_{\sigma_i+\check s}-v_{\sigma_i}$ is independent of $\frak A_{\sigma_i}$}
=\sum_{i=0}^{\infty}\bar\mu(a\Bcap b_i)\cdot\bar\mu\Bvalue{v_{\check s}>\alpha}
\Displaycause{because $v_{\sigma_i+\check s}-v_{\sigma_i}$ has the same
distribution as $v_{\check s}$}
=\bar\mu a\cdot\bar\mu\Bvalue{v_{\check s}>\alpha}.\cr}$$

\noindent Taking $a=1$, this shows that $v_{\tau+\check s}-v_{\tau}$
has the same distribution as $v_{\check s}$.   So in fact we see that

\Centerline{$\bar\mu(a\Bcap\Bvalue{v_{\tau+\check s}-v_{\tau}>\alpha})
=\bar\mu a\cdot\bar\mu\Bvalue{v_{\tau+\check s}-v_{\tau}>\alpha}$}

\noindent whenever $a\in\frak A_{\tau}$ and $\alpha\in\Bbb R$, and (using the
Monotone Class Theorem) that
$v_{\tau+\check s}-v_{\tau}$ is independent of $\frak A_{\tau}$.   Thus
$\tau\in D$.   As $\tau$ was an arbitrary member of the covered envelope of
$D$, $D$ is full.\ \Qed

\medskip

\quad{\bf (iii)} If $A\subseteq D$ is non-empty and downwards-directed, then
$\inf A\in D$.   \Prf\ Write $\tau$ for $\inf A$.   As $\Cal T_f$ is an ideal of
$\Cal T$, $\tau\in\Cal T_f$.   $A+\check s=\{\sigma+\check s:\sigma\in A\}$ is
downwards-directed (652G(b-v)) and $\inf(A+\check s)=\tau+\check s$
(652G(b-vii)).   Since $\langle\frak A_t\rangle_{t\ge 0}$ is right-continuous
and $\pmb v$ is locally near-simple,
$v_{\tau}=\lim_{\sigma\downarrow A}v_{\sigma}$ (632F),
$v_{\tau+\check s}=\lim_{\sigma\downarrow A}v_{\sigma+\check s}$
and $v_{\tau+\check s}-v_{\tau}
=\lim_{\sigma\downarrow A}v_{\sigma+\check s}-v_{\sigma}$.

Since $v_{\sigma+\check s}-v_{\sigma}$ has the same distribution as
$v_{\check s}$ for every $\sigma\in A$, $v_{\tau+\check s}-v_{\tau}$ also has
the same distribition as $v_{\check s}$, by 454Ve\Latereditions.
Since $v_{\sigma+\check s}-v_{\sigma}$  is independent of $\frak A_{\sigma}$,
and therefore of the smaller algebra $\frak A_{\tau}$. for every
$\sigma\in A$, $v_{\tau+\check s}-v_{\tau}$ is independent of $\frak A_{\tau}$,
by 652Bc.   So $\tau\in D$, as claimed.\ \Qed

\medskip

\quad{\bf (iv)} $D=\Cal T_f$.   \Prf\ We saw in (ii) that $D$ is a full sublattice of
$\Cal T_f$.   As $\check T$ separates $\Cal T_f$ (633Ea)
and is included in $D$, $D$ separates $\Cal T_f$.   If $\tau\in\Cal T_b$, then
$A_{\tau}=\{\sigma:\tau\le\sigma\in D\}$ is non-empty and downwards-directed
and has infimum $\tau$ (633Eb);  by (iii) just above, $\tau\in D$.   Thus
$D\supseteq\Cal T_b$:  as $D$ is full, $D=\Cal T_f$ (611N(e-ii)).\ \Qed

\medskip

{\bf (b)} For $C\subseteq\coint{0,\infty}$ write $\frak B_C$ for the closed
subalgebra generated by $\{v_{\tau+\check t}-v_{\tau}:t\in C\}$.

\medskip

\quad{\bf (i)} If $C\subseteq\coint{0,\infty}$ is finite then
$\frak B_C$ and $\frak A_{\tau}$ are independent.   \Prf\ Induce on $\#(C)$.
If $C$ is empty then
$\frak B_C=\{0,1\}$ and the result is trivial.   For the inductive step to
$\#(C)=n\ge 1$ enumerate $C$ in ascending order as $(t_1,\ldots,t_n)$ and set
$t_0=0$.   Then $\frak B_C$ is the closed subalgebra generated by
$\{v_{\tau+\check t_{i+1}}-v_{\tau+\check t_i}:i<n\}$.   Suppose that
$\alpha_0,\ldots,\alpha_{n-1}\in\Bbb R$ and
$a\in\frak A_{\tau}$.   Setting $a_0=a$ and
$a_{i+1}=a_i\Bcap\Bvalue{v_{\tau+\check t_{i+1}}-v_{\tau+\check t_i}>\alpha_i}$
for $i<n$, we see that $a_i\in\frak A_{\tau+\check t_i}$, while
$v_{\tau+\check t_{i+1}}-v_{\tau+\check t_i}
=v_{(\tau+\check t_i)+(t_{i+1}-t_i)\varCspcheck}-v_{\tau+\check t_i}$
is independent of $\frak A_{\tau+\check t_i}$ by (a) above,
so that $\bar\mu a_{i+1}=\bar\mu a_i
  \cdot\bar\mu\Bvalue{v_{\tau+\check t_{i+1}}-v_{\tau+\check t_i}>\alpha_i}$.
Consequently

\Centerline{$\bar\mu(a\Bcap\inf_{i<n}
   \Bvalue{v_{\tau+\check t_{i+1}}-v_{\tau+\check t_i}>\alpha_i})
=\bar\mu a\cdot\prod_{i=0}^{n-1}
   \bar\mu\Bvalue{v_{\tau+\check t_{i+1}}-v_{\tau+\check t_i}>\alpha_i}$.}

\noindent In particular,

\Centerline{$\bar\mu(\inf_{i<n}
   \Bvalue{v_{\tau+\check t_{i+1}}-v_{\tau+\check t_i}>\alpha_i})
=\prod_{i=0}^{n-1}
   \bar\mu\Bvalue{v_{\tau+\check t_{i+1}}-v_{\tau+\check t_i}>\alpha_i}$}

\noindent so

\Centerline{$\bar\mu(a\Bcap\inf_{i<n}
   \Bvalue{v_{\tau+\check t_{i+1}}-v_{\tau+\check t_i}>\alpha_i})
=\bar\mu a\cdot\bar\mu(\inf_{i<n}
   \Bvalue{v_{\tau+\check t_{i+1}}-v_{\tau+\check t_i}>\alpha_i})$.}

\noindent As $a$ and $\alpha_0,\ldots,\alpha_{n-1}$ are arbitrary,
$\frak A_{\tau}$ and $\frak B_C$ are independent
(apply 313Gb to the subsets

\Centerline{$\{\inf_{i<n}
   \Bvalue{v_{\tau+\check t_{i+1}}-v_{\tau+\check t_i}>\alpha_i}:
\alpha_0,\ldots,\alpha_{n-1}\in\Bbb R\}$,}

\Centerline{$\{d:\bar\mu(a\Bcap d)=\bar\mu a\cdot\bar\mu d$
for every $a\in\frak A_{\tau}\}$}

\noindent of $\frak A$).\ \Qed

\medskip

\quad{\bf (ii)} Now $\frak E=\bigcup\{\frak B_C:C\subseteq\coint{0,\infty}$ is
finite$\}$ is a subalgebra of $\frak B_{\coint{0,\infty}}$ and
$\bar\mu(a\Bcap e)=\bar\mu a\cdot\bar\mu e$ whenever $a\in\frak A_{\tau} $ and
$e\in\frak E$.   As $\Bcap:\frak A\times\frak A\to\frak A$ and
$\bar\mu:\frak A\to[0,1]$ are continuous for the measure-algebra
topology (323Ba, 323Ca), $\bar\mu(a\Bcap d)=\bar\mu a\cdot\bar\mu d$
whenever $a\in\frak A_{\tau}$
and $d$ belongs to the topological closure of $\frak E$, which is the closed
subalgebra generated by $\frak E$ (323J), that is, $\frak B_{\coint{0,\infty}}$.
Thus $\langle v_{\tau+\check s}-v_{\tau}\rangle_{s\ge 0}$ is independent of
$\frak A_{\tau}$.

\medskip

\quad{\bf (iii)} As for the distribution of
$\langle v_{\tau+\check s}-v_{\tau}\rangle_{s\ge 0}$, we saw in (i) of the
argument above that

$$\eqalignno{\bar\mu(\inf_{i<n}
   \Bvalue{v_{\tau+\check t_{i+1}}-v_{\tau+\check t_i}>\alpha_i})&
=\prod_{i=0}^{n-1}
   \bar\mu\Bvalue{v_{\tau+\check t_{i+1}}-v_{\tau+\check t_i}>\alpha_i}\cr&
=\prod_{i=0}^{n-1}
   \bar\mu\Bvalue{v_{(t_{i+1}-t_i)\varCspcheck}>\alpha_i}
\Displaycause{using the other clause in (a)}
=\prod_{i=0}^{n-1}
   \bar\mu\Bvalue{v_{\check t_{i+1}}-v_{\check t_i}>\alpha_i}\cr}$$

\noindent whenever $0\le t_0\le\ldots\le t_n$.   So
$\langle v_{\tau+\check s}-v_{\tau}\rangle_{s\ge 0}$ has the same
distribution as $\langle v_{\check s}\rangle_{s\ge 0}$.
}%end of proof of 652H

\leader{652I}{Theorem} Let \Asissixt\ be a right-continuous real-time stochastic
integration structure, and $\pmb v=\family{\sigma}{\Cal T_f}{v_{\sigma}}$
a L\'evy process such that
$\Osclln(\pmb v\restr[\check 0,\tau])\in L^{\infty}(\frak A)$ for every
$\tau\in\Cal T_b$.   Then $\pmb v\restr\Cal T_b$ is an $L^1$-process.

\proof{ Take $\tau^*\in\Cal T_b$;  I need to show that
$v_{\tau^*}\in L^1_{\bar\mu}$.

\medskip

{\bf (a)}  Take $t^*\ge 0$ such that $\tau^*\le\check t^*$.
Since $\pmb v$ is locally near-simple,
$\pmb v\restr\Cal T\wedge\check t^*$ is near-simple.   Set $\delta=\bover12$ and
apply the construction of 615M to $\pmb v\restr\Cal T\wedge\check t^*$
and $\delta$ to obtain families $\sequence{i}{D'_i}$, $\sequence{i}{y'_i}$ and
$\langle c'_{i\sigma}\rangle_{i\in\Bbb N,\sigma\in\Cal T\wedge\check t^*}$
as in 615M.
Set $\tau'_1=\inf D'_1$;  then $\tau'_1\in D'_1$
and $v_{\tau'_1}=y'_1$ (631Q), while $y'_0=v_{\check 0}=0$
and $c'_{0\sigma}=1$ for every $\sigma\in\Cal T\wedge\check t^*$.

Since $\tau'_1\in D'_1$, the formula in 615Ma tells us that
$\Bvalue{\tau'_1<\check t^*}\Bsubseteq\Bvalue{|v_{\tau'_1}-0|\ge 1}$;  so
if $\tau'_1=\check 0$ this must be because $\check t^*=\check 0$,
$\tau^*=\check 0$,
$v_{\tau^*}=0\in L^1_{\bar\mu}$ and we're done.   So
suppose from now on that $\tau'_1\ne\check 0$.   In this case there is an
integer $m\ge 1$ such that
$\eta\in\ocint{0,t^*}$ such that $a=\Bvalue{\tau'_1>\eta}$ is non-zero,
where $\eta=\bover{t^*}m$.
If $0\le s\le\eta$ and we set $\sigma=\check s\wedge\tau'_1$, then

$$\eqalignno{a&
=\Bvalue{\tau'_1>\eta}\Bsetminus\Bvalue{\check s>\eta}
\Bsubseteq\Bvalue{\check s<\tau'_1}
\Bsubseteq\Bvalue{\sigma<\tau'_1}
=c'_{0\sigma}\Bsetminus c'_{1\sigma}
\Bsubseteq\Bvalue{|v_{\sigma}|<\bover12}
\Displaycause{615M(d-v)}
\Bsubseteq\Bvalue{|v_{\check s}|<\bover12}\Bcup\Bvalue{\tau'_1<\check s}\cr}$$

\noindent so in fact $a\Bsubseteq\Bvalue{|v_{\check s}|<\bover12}$.

\medskip

{\bf (b)} Now re-apply 615M to $\pmb v\restr\Cal T\wedge\check t^*$, this time
with $\delta=1$, to get $\sequence{i}{D_i}$, $\sequence{i}{y_i}$ and
$\langle c_{i\sigma}\rangle_{i\in\Bbb N,\sigma\in\Cal T\wedge\check t^*}$.
Set $\tau_i=\inf D_i$ for $i\in\Bbb N$;  this time we have $\tau_i\in D_i$ so
$c_{i\sigma}=\Bvalue{\tau_i\le\sigma}$
for $i\in\Bbb N$ and $\sigma\le\check t^*$.

For $i\in\Bbb N$, set
$a_i=\inf_{0\le s\le\eta}\Bvalue{|v_{\tau_i+\check s}-v_{\tau_i}|<\bover12}$, so
that $\bar\mu a_0\ge\bar\mu a$ is non-zero.   Since
$\langle v_{\tau_i+\check s}-v_{\tau_i}\rangle_{s\ge 0}$ has the same
distribution as $\langle v_{\check s}\rangle_{s\ge 0}$ (652Hb),
$\bar\mu a_i=\bar\mu a_0$ for every $i\ge n$;  write $\gamma$ for this common
value.   Since
$\langle v_{\tau_i+\check s}-v_{\tau_i}\rangle_{s\ge 0}$ is independent of
$\frak A_{\tau_i}$ for each $i$, $\bar\mu(a_i\Bcap b)=\gamma\bar\mu b$ for every
$i\in\Bbb N$ and $b\in\frak A_{\tau_i}$;  as also $a_i\in\frak A_{\tau_{i+1}}$
for every $i$, $\sequence{i}{a_i}$ is independent.

\medskip

{\bf (c)} The point is that
$a_i\Bcap\Bvalue{\tau_{i+1}<\check t^*}
\Bsubseteq\Bvalue{\tau_i+\check\eta\le\tau_{i+1}}$
for every $i\in\Bbb N$.   \Prf\ We know that $\{\tau_i+\check s:s\ge 0\}$
separates $\Cal T\vee\tau_i$ (652G(b-vi)), so
$A=\{\tau_i+\check s:0\le s\le\eta\}$ separates
$\Cal T\cap[\tau_i,\tau_i+\check\eta]$ (633C(b-iii)).
Now $a_i\Bsubseteq\Bvalue{|v_{\sigma}-v_{\tau_i}|\le\bover12}$ whenever
$\sigma\in A$.   If $\sigma$ belongs to the covered envelope $\hat A$ of $A$,
then

\Centerline{$a_i
=\sup_{\rho\in A}a_i\Bcap\Bvalue{\sigma=\rho}
    \Bcap\Bvalue{|v_{\rho}-v_{\tau_i}|\le\bover12}
\Bsubseteq\Bvalue{|v_{\sigma}-v_{\tau_i}|\le\bover12}$.}

\noindent Now consider $\tau=(\tau_i+\check\eta)\wedge\tau_{i+1}$ and
$\hat A_{\tau}=\{\sigma:\tau\le\sigma\in\hat A\}$.   Then $\hat A_{\tau}$ is
downwards-directed and its infimum is $\tau$, by 633Eb, so
$v_{\tau}=\lim_{\sigma\downarrow\hat A_{\tau}}v_{\sigma}$ (632F),

\Centerline{$|v_{\tau}-v_{\tau_i}|\times\chi a_i
=\lim_{\sigma\downarrow\hat A_{\tau}}|v_{\sigma}-v_{\tau_i}|\times\chi a_i
\le\bover12\chi a_i$}

\noindent and $a_i\Bsubseteq\Bvalue{|v_{\tau}-v_{\tau_i}|\le\bover12}$.
On the other hand,
$\Bvalue{\tau_{i+1}<\check t^*}
\Bsubseteq\Bvalue{|v_{\tau_{i+1}}-v_{\tau_i}|\ge 1}$ so

$$\eqalign{a_i\Bcap\Bvalue{\tau_{i+1}<\check t^*}
     \Bcap\Bvalue{\tau_{i+1}<\tau_i+\check\eta}
&\Bsubseteq\Bvalue{|v_{\tau}-v_{\tau_i}|\le\bover12}
     \Bcap\Bvalue{|v_{\tau_{i+1}}-v_{\tau_i}|\ge 1}
     \Bcap\Bvalue{\tau_{i+1}=\tau}\cr&
\Bsubseteq\Bvalue{|v_{\tau}-v_{\tau_i}|\le\bover12}
     \Bcap\Bvalue{|v_{\tau}-v_{\tau_i}|\ge 1}
=0\cr}$$

\noindent and
$a_i\Bcap\Bvalue{\tau_{i+1}<\check t^*}
\Bsubseteq\Bvalue{\tau_i+\check\eta\le\tau_{i+1}}$.\ \Qed

\medskip

{\bf (d)} If $I\subseteq n\in\Bbb N$ then
$\inf_{i\in I}a_i\Bcap\Bvalue{\tau_n<\check t^*}
\Bsubseteq\Bvalue{(\eta\#(I))\varCspcheck\le\tau_n}$.   \Prf\ Induce on
$n$.   For $n=0$ this is trivial.   For the inductive step to $n+1$, if
$I\subseteq n+1$ and $n\notin I$ then $I\subseteq n$ and

$$\eqalign{\inf_{i\in I}a_i\Bcap\Bvalue{\tau_{n+1}<\check t^*}
&\Bsubseteq\inf_{i\in I}a_i\Bcap\Bvalue{\tau_n<\check t^*}\cr&
\Bsubseteq\Bvalue{(\eta\#(I))\varCspcheck\le\tau_n}
\Bsubseteq\Bvalue{(\eta\#(I))\varCspcheck\le\tau_{n+1}}.\cr}$$

\noindent If $n\in I$ set $J=I\setminus\{n\}$;  then

$$\eqalignno{\inf_{i\in I}a_i\Bcap\Bvalue{\tau_{n+1}<\check t^*}
&=a_n\Bcap\Bvalue{\tau_{n+1}<\check t^*}
    \Bcap\inf_{i\in J}a_i\Bcap\Bvalue{\tau_n<\check t^*}\cr&
\Bsubseteq\Bvalue{\tau_n+\check\eta\le\tau_{n+1}}
    \Bcap\Bvalue{(\eta\#(J))\varCspcheck\le\tau_n}
\Displaycause{by (c) above and the inductive hypothesis}
\Bsubseteq\Bvalue{(\eta\#(J))\varCspcheck+\check\eta\le\tau_{n+1}}
\Displaycause{652G(b-v)}
=\Bvalue{(\eta\#(I))\varCspcheck\le\tau_{n+1}}\cr}$$

\noindent by 652G(b-iv).   Thus the induction continues.\ \Qed

\medskip

{\bf (e)} Now $\bar\mu\Bvalue{\tau_n<\tau^*}\le n^m(1-\gamma)^{n-m}$
for every $n>m$.   \Prf\ If $I\subseteq n$ and $\#(I)>m$, then

$$\eqalignno{\Bvalue{\tau_n<\tau^*}\Bcap\inf_{i\in I}a_i&
\Bsubseteq\Bvalue{\tau_n<\check t^*}\Bcap\inf_{i\in I}a_i\cr&
\Bsubseteq\Bvalue{(\eta\#(I))\varCspcheck\le\tau_n}
    \Bcap\Bvalue{\tau_n<\check t^*}
\Displaycause{by (c)}
=0\cr}$$

\noindent because $\eta\#(I)\ge t^*$.    So

$$\eqalign{\Bvalue{\tau_n<\tau^*}&
=\sup_{I\subseteq n}(\Bvalue{\tau_n<\tau^*}
   \Bcap\inf_{i\in I}a_i
   \Bcap\inf_{i\in n\setminus I}1\Bsetminus a_i)\cr&
\Bsubseteq\sup_{I\subseteq n,\#(I)\le m}
   \inf_{i\in n\setminus I}1\Bsetminus a_i)\cr}$$

\noindent has measure at most $n^m(1-\gamma)^{n-m}$, since
the $a_i$ are independent and $\bar\mu(1\Bsetminus a_i)=1-\gamma$
for every $i$, by (b).\ \Qed

\medskip

{\bf (f)} Everything I have said so far applies to any L\'evy process.   But
now consider the hypothesis on the oscillations of
$\pmb v\restr[\check 0,\tau]$ for $\tau\in\Cal T_b$.   Let $\beta\ge 0$ be such
that $\Osclln(\pmb v\restr[\check 0,\check t^*])\le\beta\chi 1$.
Then $|y_{i+1}-y_i|\le(\beta+1)\chi 1$ for every $i\in\Bbb N$ (618N), so
$|y_n|\le n(\beta+1)\chi 1$ for every $n\in\Bbb N$, and

$$\eqalignno{\Bvalue{\tau^*<\tau_n}&
=\sup_{i<n}\Bvalue{\tau_i\le\tau^*}\Bsetminus\Bvalue{\tau_{i+1}\le\tau^*}
=\sup_{i<n}c_{i\tau^*}\Bsetminus c_{i+1,\tau^*}
\Bsubseteq\sup_{i<n}\Bvalue{|v_{\tau^*}-y_i|<1}
\Displaycause{615Md}
\Bsubseteq\Bvalue{|v_{\tau^*}\le n(\beta+1)};\cr}$$

\noindent since also

\Centerline{$\Bvalue{\tau^*=\tau_n}\Bsubseteq\Bvalue{v_{\tau^*}=y_n}
\Bsubseteq\Bvalue{|v_{\tau^*}\le n(\beta+1)}$,}

\noindent
$\Bvalue{|v_{\tau^*}>n(\beta+1)}\Bsubseteq\Bvalue{\tau_n<\tau^*}$
has measure at most $n^m(1-\gamma)^{n-m}$ for $n>m$.   But $\gamma=\bar\mu a_0$
is greater than $0$, so

$$\eqalignno{\|v_{\tau^*}\|_1&
\le\sum_{n=0}^{\infty}(\beta+1)\bar\mu\Bvalue{|v_{\tau^*}|>n(\beta+1)}
\Displaycause{365A}
\le (m+1)(\beta+1)+\sum_{n=m+1}^{\infty}n^m(1-\gamma)^{n-m}\cr}$$

\noindent is finite, and $v_{\tau^*}\in L^1_{\bar\mu}$, as required.
}%end of proof of 652I

\leader{652J}{Proposition} Let \Asissixt\ be a right-continuous
real-time stochastic
integration structure, and $\pmb v=\family{\sigma}{\Cal T_f}{v_{\sigma}}$
a L\'evy process such that $\pmb v\restr\Cal T_b$ is an $L^1$-process.
Then there is an $\alpha\in\Bbb R$ such that
$\pmb v-\alpha\pmb\iota$ is a local martingale.

\proof{{\bf (a)} Consider first the case in which $\Expn(v_{\check t})=0$ for
every $t\ge 0$.

\medskip

\quad{\bf (i)} If $0\le s\le t$, then $v_{\check t}-v_{\check s}$ is
independent of $\frak A_s$ and has zero expectation;  so
$\Expn(\chi b\times(v_{\check t}-v_{\check s})
=\bar\mu b\cdot\Expn(v_{\check t}-v_{\check s})=0$ for every $b\in\frak A_s$.
Writing $P_{\check s}$ for the conditional expectation associated with the
closed subalgebra $\frak A_{\check s}=\frak A_s$,
$P_{\check s}(v_{\check t}-v_{\check s})=0$ and
$P_{\check s}(v_{\check t})=P_{\check s}(v_{\check s})=v_{\check s}$.
Thus $\pmb v\restr\coint{0,\infty}\varCspcheck$ is a martingale.

\medskip

\quad{\bf (ii)} Now there is a local martingale $\pmb w$ with domain $\Cal T_f$
extending $\pmb  v\restr\coint{0,\infty}\varCspcheck$ (622Ob), and
$\pmb w$ is locally near-simple (632I).   Because $\pmb v$ and $\pmb w$ are
both locally near-simple and agree on $\coint{0,\infty}\varCspcheck$, they
are equal (633F again).   So $\pmb v$ is a local martingale.

\medskip

{\bf (b)(i)} For the general case, set $f(t)=\Expn(v_{\check t})$ for $t\ge 0$.
Then $f(s+t)=f(s)+f(t)$ for all $s$, $t\ge 0$, because
$f(s+t)-f(t)=\Expn(v_{(s+t)}\varCspcheck=v_{\check t})$
and $v_{(s+t}\varCspcheck=v_{\check t}$ has the same distribution as
$v_{\check s}$ and therefore has the same expectation.   As
$f(\bover1n)=\bover1nf(1)$ for every integer $n\ge 1$, and
$v_{\check 0}=0$, $f(q)=qf(1)$ for every rational $q\ge 0$.

\medskip

\quad{\bf (ii)} $f:\coint{0,\infty}\to\Bbb R$ is Borel measurable.   \Prf\ For
$M$, $t\ge 0$, set $f_M(t)=\Expn(\med(-M\chi 1,v_{\check t},M\chi 1))$.
We know
that $\lim_{t\downarrow 0}v_{\check t}=0$ for the topology of convergence in
measure (652D), so

\Centerline{$\lim_{t\downarrow s}v_{\check t}-v_{\check s}
=\lim_{t\downarrow s}v_{\check t}-v_{\check s}
=\lim_{t\downarrow s}v_{(t-s)\varCspcheck}=0$}

\noindent for every $s\ge 0$;  similarly,

\Centerline{$\lim_{t\uparrow s}v_{\check t}-v_{\check s}
=-\lim_{t\uparrow s}v_{\check s}-v_{\check t}
=-\lim_{t\uparrow s}v_{(s-t)\varCspcheck}=0$}

\noindent for every $s>0$.   Thus
$t\mapsto v_{\check t}:\coint{0,\infty}\to L^0$ is continuous, and it
follows that $t\mapsto\med(-M\chi 1,v_{\check t},M\chi 1)$ is continuous for the
topology of convergence in measure.   But this agrees with the norm topology of
$L^1_{\bar\mu}$ on the uniformly integrable set $[-M\chi 1,M\chi 1]$, so
$t\mapsto\med(-M\chi 1,v_{\check t},M\chi 1)$ is $\|\,\|_1$-continuous and
$f_M:\coint{0,\infty}\to\Bbb R$ is continuous.   Now
$f(t)=\lim_{n\to\infty}f_n(t)$ for every $t$, so $f$ is Borel measurable.\ \Qed

\medskip

\quad{\bf (iii)} Consequently there is a $\delta>0$ such that $f$ is bounded on
$[0,\delta]$.  \Prf\ There is a compact set $K\subseteq\coint{0,\infty}$, of
non-zero Lebesgue measure, such that
$f\restr K$ is continuous, therefore bounded.   Now
$f\restr(K-K)\cap\coint{0,\infty}$ is bounded and
$K-K$ includes
$[-\delta,\delta]$ for some $\delta>0$ (443Dc), so $f\restr[0,\delta]$ is
bounded.\ \Qed

\medskip

\quad{\bf (iv)} $\lim_{t\downarrow 0}f(t)=0$.   \Prf\ Set
$M=\sup_{t\in[0,\delta]}|f(t)|$.   If $n\in\Bbb N$ and $0\le t\le 2^{-n}\delta$,
$M\ge|f(2^nt)|=2^n|f(t)|$ and $|f(t)|\le 2^{-n}M$.\ \QeD\   Since
$|f(s)-f(t)|=|f(|s-t|)|$ for all $s$, $t\ge 0$, $f$ is continuous.   Now
$f(q)=qf(1)$ for every rational $q>0$, so $f(t)=tf(1)$ for every
$t\ge 0$.

\medskip

\quad{\bf (v)} Now consider $\pmb w=\pmb v-f(1)\pmb\iota$.   Because both
$\pmb v$ and $\pmb\iota$ are locally near-simple with domain $\Cal T_f$ (631Ea),
so is $\pmb w$.   Express $\pmb w$ as $\family{\sigma}{\Cal T_f}{w_{\sigma}}$.
If $s$, $t\ge 0$, the distribution of
$w_{(s+t)\varCspcheck}-w_{\check s}
=v_{(s+t)\varCspcheck}-v_{\check s}+tf(1)\chi 1$
is the same as that of $w_{\check t}$, and
$w_{(s+t)\varCspcheck}-w_{\check s}$ is independent of $\frak A_s$,
so $\pmb w$ is a L\'evy process.   For $\sigma\in\Cal T_b$,
$\iota_{\sigma}\in L^{\infty}(\frak A)\subseteq L^1_{\bar\mu}$, so
$w_{\sigma}\in L^1_{\bar\mu}$;  while $\Expn(w_{\check s})=f(s)-f(1)s=0$ for
every $s\ge 0$.   By (a), $\pmb w$ is a local martingale, and we have an
expression of the right form for $\pmb v$.
}%end of proof of 652J

\leader{652K}{Theorem} Let \Asissixt\ be a right-continuous
real-time stochastic
integration structure, and $\pmb v=\family{\sigma}{\Cal T_f}{v_{\sigma}}$
a L\'evy process.   Then $\pmb v$ is a \sm, therefore a local integrator.

\proof{{\bf (a)}
Set $\psi(\sigma,\tau)=\med(-\chi 1,v_{\tau}-v_{\sigma},\chi 1)$ for
$(\sigma,\tau)\in\Cal T_f^{2\uparrow}$.   Applying 633M(b-i) to
$\psi\restr(\Cal T\wedge\tau)^{2\uparrow}$ for $\tau\in\Cal T_f$, we see that
$\psi$ is a strictly adapted interval function.   By 633Mc,

\Centerline{$\pmb w=ii_{\psi}(\tbf 1)
=\family{\tau}{\Cal T_f}{\int_{\Cal T\wedge\tau}d\psi}$}

\noindent is defined everywhere in $\Cal T_f$;  by 633Mg, $\pmb w$ is locally
near-simple;  by 633Me, $\Osclln(\pmb w\restr\Cal T\wedge\tau)\le\chi 1$ for
every $\tau\in\Cal T_f$.   By 633Md, $\pmb v-\pmb w$ is locally of bounded
variation.

\medskip

{\bf (b)} The point is that $\pmb w$ is a L\'evy process.   \Prf\ Set
$\psi_0=\psi\restr(\coint{0,\infty}\varCspcheck)^{2\uparrow}$.   Since
$\pmb v\restr\coint{0,\infty}\varCspcheck$ is moderately oscillatory,
%actually near-simple I suppose
$\pmb w_0=ii_{\psi_0}(\tbf 1)$ is defined everywhere on
$\coint{0,\infty}\varCspcheck$, and by 633N (applied to
$\psi\restr(\Cal T\wedge\check t)^{2\uparrow}$ for $t\ge 0$)
$\pmb w_0=\pmb w\restr\coint{0,\infty}\varCspcheck$.

Express $w$ as $\family{\sigma}{\Cal T_f}{w_{\sigma}}$.   If $s$, $t\ge 0$,
then $\langle v_{(s+t)\varCspcheck}v_{\check s}\rangle_{t\ge 0}$ is independent
of $\frak A_s$ (652Hb) and has the same distribution as
$\langle v_{\check t}\rangle_{t\ge 0}$.   Set

$$\eqalign{\psi'(\check t,\check t')&
=\psi((s+t)\varCspcheck,(s+t')\varCspcheck))
=\med(-\chi 1,v_{(s+t)\varCspcheck}-v_{(s+t')\varCspcheck},\chi 1)\cr&
=\med(-\chi 1,
    (v_{(s+t)\varCspcheck}-v_{\check s})-(v_{(s+t')\varCspcheck}-v_{\check s}),
    \chi 1)\cr}$$

\noindent for $0\le t\le t'$;  then $\psi'$ and $\psi_0$ have the same
distribution, that is,

\Centerline{$
\bar\mu(\inf_{i\le n}\Bvalue{\psi'(\check t_i,\check t_i')>\alpha_i})
=\bar\mu(\inf_{i\le n}\Bvalue{\psi_0(\check t_i,\check t_i')>\alpha_i})$}

\noindent whenever $0\le t_i\le t'_i$ and $\alpha_i\in\Bbb R$ for $i\le n$
(454Vd\Latereditions).
At the same time, $\psi'$ is independent of $\frak A_s$, that is,

\Centerline{$
\bar\mu(a\Bcap\inf_{i\le n}\Bvalue{\psi'(\check t_i,\check t_i')>\alpha_i})
=\bar\mu a
   \cdot\bar\mu(\inf_{i\le n}\Bvalue{\psi'(\check t_i,\check t_i')>\alpha_i})$}

\noindent whenever $a\in\frak A_s$ and
$0\le t_i\le t'_i$ and $\alpha_i\in\Bbb R$ for $i\le n$.   It follows that
$I\mapsto S_I(\psi',\tbf 1):\coint{0,\infty}^{<\omega}\to L^0$ and
$I\mapsto S_I(\psi_0,\tbf 1)$ have the same distribution, that is,

\Centerline{$\bar\mu(\inf_{i\le n}\Bvalue{S_{I_i}(\psi',\tbf 1)>\alpha_i})
=\bar\mu(\inf_{i\le n}\Bvalue{S_{I_i}(\psi_0,\tbf 1)>\alpha_i})$}

\noindent whenever $I_0,\ldots,I_n$ are finite subsets of $\{\check t:t\ge 0\}$
and $\alpha_i\in\Bbb R$ for $i\le n$, while
$I\mapsto S_I(\psi',\tbf 1)$ is independent of $\frak A_s$.   Now this means
that if $t\ge 0$ then the limits

\Centerline{$\int_{\coint{0,\infty}\varCspcheck\wedge\check t}d\psi'
=\int_{\coint{0,\infty}\varCspcheck\cap[\check s,(s+t)\varCspcheck)}d\psi
=w_{(s+t)\varCspcheck}-w_{\check s}$}

\noindent and

\Centerline{$\int_{\coint{0,\infty}\varCspcheck\wedge\check t}d\psi_0
=w_{\check t}$}

\noindent have the same distribution (454Ve again), while
$w_{(s+t)\varCspcheck}-w_{\check s}$ is independent of $\frak A_s$
(652Bc).   As $s$ and $t$ are arbitrary, $\pmb w$ is a L\'evy process.\ \Qed

\medskip

{\bf (c)} Since $\Osclln(\pmb w\restr\Cal T\wedge\tau)\le\chi 1$ for
every $\tau\in\Cal T_f$, $\pmb w\restr\Cal T_b$ is an $L^1$-process (652I) and
there is an $\alpha\in\Bbb R$ such that $\pmb w-\alpha\pmb\iota$ is a local
martingale (652J).   But now our original process

\Centerline{$\pmb v=(\pmb v-\pmb w)+\alpha\pmb\iota+(\pmb w-\alpha\pmb\iota)$}

\noindent is expressible as the sum of a locally order-bounded process and a
local martingale, so is a \sm, that is, is a local integrator (627Q).
}%end of proof of 652K

\leader{652L}{Proposition} Let \Asissixt\ be a right-continuous
real-time stochastic integration structure, and
$\pmb v=\family{\sigma}{\Cal T_f}{v_{\sigma}}$ a L\'evy process.
Then its quadratic variation $\pmb v^*$ is a L\'evy process.

\proof{ Since $\pmb v$ is a local integrator (652K), its quadratic variation is
defined everywhere in $\Cal T_b$.   Express $\pmb v^*$ as
$\family{\tau}{\Cal T_b}{v^*_{\tau}}$.

\medskip

{\bf (a)} By 631Jb, $\pmb v^*$ is locally near-simple.

\medskip

{\bf (b)} We know that $\check T$ separates $\Cal T_b$ (633Da), so if $t\ge 0$
then $\check T\wedge\check t=\{\check s:0\le s\le t\}$ separates
$\Cal T_b\wedge\check t$ (633D(b-i)) and contains both
$\min(\Cal T_b\wedge\check t)=\check 0$
and $\max(\Cal T_b\wedge\check t)=\check t$.   Also
$\pmb v\restr\check T\wedge\check t$ is an
integrator (616P(b-ii)).   By 633Ph, $\pmb v^*$ extends the quadratic variation
of $\pmb v\restr\check T\wedge\check t$;  in particular,

\Centerline{$v^*_{\check t}=\int_{\check T\wedge\check t}(d\pmb v)^2
=\lim_{I\uparrow\Cal I(\check T\wedge\check t)}S_I(\tbf 1,(d\pmb v)^2)$.}

\medskip

{\bf (c)} Now if $s$, $t\ge 0$ and
$I\in\Cal I(\check T\wedge\check t)=(\check T\wedge\check t)^{<\omega}$,
then $S_{I+\check s}(\tbf 1,(d\pmb v)^2)$ is independent of $\frak A_s$ and
has the same distribution as
$S_I(\tbf 1,(d\pmb v)^2)$.   \Prf\
If $\#(I)\le 1$, then $S_{I+\check s}(\tbf 1,(d\pmb v)^2)$ and
$S_I(\tbf 1,(d\pmb v)^2)$ are both zero.
Otherwise, let $\langle s_i\rangle_{i\le n}$ be the increasing enumeration of
$I$;  then $\langle v_{(s+s_i)\varCspcheck}\rangle_{i\le n}$ is independent of
$\frak A_s$ and has the same distribution as
$\langle v_{\check s_i}\rangle_{i\le n}$, by 652Hb, so
$S_{I+\check s}(\tbf 1,(d\pmb v)^2)
=\sum_{i=0}^{n-1}(v_{(s+s_{i+1})\varCspcheck}-v_{(s+s_i)\varCspcheck})^2$ is
independent of $\frak A_s$ and
has the same distribution as
$\sum_{i=0}^{n-1}(v_{\check s_{i+1}}-v_{\check s_i})^2
=S_{I+\check s}(\tbf 1,(d\pmb v)^2)$.\ \QeD

\medskip

{\bf (d)}  Consequently

$$\eqalign{v^*_{(s+t)\varCspcheck}-v^*_{\check s}&
=\int_{\check T\wedge(s+t)\varCspcheck}(d\pmb v)^2
   -\int_{\check T\wedge\check s}(d\pmb v)^2
=\int_{\check T\cap[\check s,(s+t)\varCspcheck]}(d\pmb v)^2\cr&
=\lim_{I\uparrow\Cal I(\check T\cap[\check s,(s+t)\varCspcheck])}
   S_I(\tbf 1,(d\pmb v)^2)
=\lim_{I\uparrow\Cal I(\check T\wedge\check t)}
   S_{I+\check s}(\tbf 1,(d\pmb v)^2)\cr}$$

\noindent is independent of $\frak A_s$, by 652Bc, and
has the same distribution as
$\int_{\check T\wedge\check t}(d\pmb v)^2=v^*_{\check t}$ by 454Ve.
Thus $\pmb v^*$ satisfies the conditions of 652C and is a L\'evy process.
}%end of proof of 652L

\leader{652M}{The Cauchy process} \cmmnt{I mentioned the Cauchy distribution
in the exercises to \S285, %285Xp
but I wish now to go rather more deeply
into its properties, so I run over the basic facts from the beginning.

\medskip

}{\bf (a)} If $t>0$, then\prooflet{

\Centerline{$\int_{-\infty}^{\infty}\Bover{t}{\pi(t^2+\xi^2)}d\xi
=\Bover1{\pi}\int_{-\infty}^{\infty}\Bover{d}{d\xi}\arctan(1+\bover{\xi}t)d\xi
=1$,}

\noindent so} we have a distribution $\lambda_t$ on $\Bbb R$ with
density function $\xi\mapsto\bover{t}{\pi(t^2+\xi^2)}$\prooflet{ (271H)}, the
{\bf Cauchy distribution} with centre $0$ and scale parameter $t$.

\spheader 652Mb{\bf (i)} For $t>0$\prooflet{ and $\eta\in\Bbb R$,

\Centerline{$\int_0^{\infty}e^{-t\xi}\cos\eta\xi\,d\xi
=\Bover1{t^2+\eta^2}\int_0^{\infty}\Bover{d}{d\xi}
   (\eta e^{-t\xi}\sin\eta\xi-te^{-t\xi}\cos\eta\xi)d\xi
=\Bover{t}{t^2+\eta^2}$.}

\noindent So

\Centerline{$\int_{-\infty}^{\infty}e^{-i\eta\xi}e^{-t|\xi|}d\xi
=\int_{-\infty}^{\infty}e^{-t|\xi|}\cos\eta\xi\,d\xi
=\Bover{2t}{t^2+\eta^2}$,}

\noindent the Fourier transform of $\xi\mapsto e^{-t|\xi|}$ is
$\eta\mapsto\Bover{2t}{\sqrt{2\pi}(t^2+\eta^2)}$ and the inverse Fourier
transform of $\eta\mapsto\Bover{2t}{\sqrt{2\pi}(t^2+\eta^2)}$
is $\xi\mapsto e^{-t|\xi|}$ (283J), that is,

\Centerline{$e^{-t|\xi|}
=\int_{-\infty}^{\infty}
   e^{i\xi\eta}\Bover{t}{\pi(t^2+\eta^2)}d\eta
=\int_{-\infty}^{\infty}
   e^{i\xi\eta}\lambda_t(d\eta)$}

\noindent for $\xi\in\Bbb R$ (235K), and} the characteristic function of
$\lambda_t$ is $\xi\mapsto e^{-t|\xi|}$\prooflet{ (285Aa)}.

\medskip

\quad{\bf (ii)}\cmmnt{ The argument gives us another integral:}
$\biggerint_0^{\infty}\Bover{1-\cos\eta}{\eta^2}d\eta=\Bover{\pi}{2}$.
\prooflet{\Prf\ For any $t>0$,

\Centerline{$\int_0^{\infty}\Bover{\cos\eta}{t^2+\eta^2}d\eta
=\Bover12\int_{-\infty}^{\infty}\Bover{e^{i\eta}}{t^2+\eta^2}d\eta
=\Bover{\pi}{2t}e^{-t}$}

\noindent and

\Centerline{$\int_0^{\infty}\Bover1{t^2+\eta^2}d\eta
=\Bover1t\int_0^{\infty}\Bover1{1+\eta^2}d\eta
=\Bover{\pi}{2t}$.}

\noindent Because $\cos\eta\le 1$, $\sin\eta\le\eta$,
$1-\cos\eta\le\bover12\eta^2$ and
$0\le\Bover{1-\cos\eta}{\eta^2}\le\min(\Bover12,\Bover1{\eta^2})$
for every $\eta>0$,
$\biggerint_0^{\infty}\Bover{1-\cos\eta}{\eta^2}d\eta$
is defined in $\Bbb R$ and equal to

\Centerline{$\lim_{t\downarrow 0}\int_0^{\infty}
   \Bover{1-\cos\eta}{t^2+\eta^2}d\eta
=\lim_{t\downarrow 0}\Bover{\pi}{2t}(1-e^{-t})=\Bover{\pi}2$.  \Qed}}

\spheader 652Mc If $s$, $t>0$ \prooflet{the
characteristic function of $\lambda_s*\lambda_t$ is
$\xi\mapsto e^{-s|\xi|}e^{-t|\xi|}=e^{-(s+t)|\xi|}$ (285R), that is, it is equal
to the characteristic function of $\lambda_{s+t}$, and}
$\lambda_s*\lambda_t=\lambda_{s+t}$\prooflet{ (285Ma)}.   \cmmnt{

}We can\cmmnt{ therefore} apply the construction of 652F with the family
$\langle\lambda_t\rangle_{t>0}$ to obtain a probability space
$(\Cdlg,\ddot\Sigma,\ddot\mu)$, a stochastic integration structure
$(\frak C,\bar{\ddot\mu},\coint{0,\infty},\langle\frak C_t\rangle_{t\ge 0})$ and
a classical L\'evy process $\pmb u$, the {\bf Cauchy process}.

\spheader 652Md Writing $C(\coint{0,\infty})\subseteq\Cdlg$
for the set of continuous functions from $\coint{0,\infty}$ to $\Bbb R$,
$\ddot\mu C(\coint{0,\infty})=0$.
\prooflet{\Prf\ Writing $X_t(\omega)=\omega(t)$
for $\omega\in\Cdlg$ and $t\ge 0$ as usual,

\Centerline{$\ddot\mu\{\omega:|X_t(\omega)|\le\epsilon\}
=1-2\int_{\epsilon}^{\infty}\Bover{t}{\pi(t^2+\xi^2)}d\xi
\le 1-\Bover2{\pi}\int_{\epsilon}^{\infty}\Bover{t}{2\xi^2}d\xi
=1-\Bover{t}{\pi\epsilon}$}

\noindent whenever $0<t\le\epsilon$.   So if $n>\bover1{\epsilon}$ is an
integer and we set
$F_n(\epsilon)=\{\omega:|X_{t_{i+1}}(\omega)-X_{t_i}(\omega)|\le\epsilon
\text{ for every }i<n\}$ where $t_i=\bover{i}n$ for $i\le n$,

\Centerline{$\ddot\mu F_n(\epsilon)
=\ddot\mu\{\omega:|X_{t_1}|\le\epsilon\}^n
\le(1-\Bover1{n\pi\epsilon})^n
\le e^{-1/\pi\epsilon}$}

\noindent because the $X_{t_{i+1}}-X_{t_i}$ are independent with the same
distribution as $X_{t_1}$ and
$\ln(1-\bover1{n\pi\epsilon})\le-\bover1{n\pi\epsilon}$.
But now

\Centerline{$C(\coint{0,\infty})
\subseteq\bigcup_{m\in\Bbb N}\bigcap_{n\ge m}F_n(\epsilon)$}

\noindent has measure at most
$\liminf_{n\to\infty}\ddot\mu F_n(\epsilon)\le e^{-1/\pi\epsilon}$.
As $\epsilon$ is
arbitrary, $C(\coint{0,\infty})$ is negligible.\ \Qed}

\leaveitout{Note that $\lambda_tE=\lambda_1(\bover1tE)$ for every Borel set
$E\subseteq\Bbb R$ and $t>0$.   \Prf\ Apply 235A with the substitution
$\phi(\xi)=t\xi$ in the formula
$\lambda_tE=\int_E\Bover{t}{\pi(t^2+\xi^2)}d\xi$.\ \Qed

\spheader 652M?
 $t(\bover{\pi}2-\arctan t)=t\int_t^{\infty}\Bover1{1+\xi^2}d\xi
\le t\int_{t}^{\infty}\Bover1{\xi^2}d\xi=1$ for every
$t>0$.

\spheader 652M?
Suppose that $(\frak A,\bar\mu)$ is a probability algebra and
$v\in L^0$ has distribution $\lambda_t$ where $t>0$.
Then $\|\med(-M\chi 1,v,M\chi 1)\|_2^2\le\Bover{4Mt}{\pi}$ for any $M\ge 0$.
\prooflet{\Prf ... \Qed}
}

\dvro{\leader{652N}{Alternative description of the Cauchy process}
Let $\ddot\mu$ be the probability measure on $\Cdlg$ defined in 652M.
Let $\mu_0$ be the indefinite-integral
measure over $\mu_L$ corresponding to the function
$\xi\mapsto\Bover1{\pi\xi^2}:\BbbR\setminus\{0\}\to\coint{0,\infty}$%
\cmmnt{ (234J)},
$\mu_1$ the subspace measure on $\coint{0,\infty}$ induced by $\mu_L$, and
$\mu=\mu_0\times\mu_1$ the c.l.d.\ product measure on
$S=\Bbb R\times\coint{0,\infty}$\cmmnt{ (251F)}.
Let $\nu$ be the Poisson point process on $(S,\mu)$ with intensity $1$, and
$\ddot\mu$ the measure on $\Cdlg$ described in 652Mc.
For $\omega\in\Cdlg$ write $\Jump(\omega)$ for

\Centerline{$\{(\xi,t):\xi\in\Bbb R\setminus\{0\}$, $t>0$,
$\omega(t)=\lim_{s\uparrow t}\omega(s)+\xi\}$.}

\noindent Then $\Jump:\Cdlg\to\Cal PS$ is \imp\ for $\ddot\mu$ and $\nu$.
If $E\subseteq S$ is such that $\mu E$ is defined and finite
then $\ddot\mu\{\omega:E\cap\Jump(\omega)=\emptyset\}=e^{-\mu E}$;  in
particular, if $t\ge 0$ and $\alpha>0$, then

\Centerline{$\ddot\mu\{\omega:
    \Jump(\omega)\cap(\coint{\alpha,\infty}\times[0,t])\ne\emptyset\}
=1-e^{-t/\alpha}$.}

\noindent Suppose that $\mu E$ is defined and finite, and
$(-\xi,s)\in E$ whenever $(\xi,s)\in E$.   Setting
$X_E(\omega)=\sum_{(\xi,s)\in E\cap\Jump(\omega)}\xi$,
$\Expn(X_E)=0$ and $\Expn(X_E^2)=\int_E\xi^2\mu(d\xi)$ is finite.

For $\ddot\mu$-almost every $\omega\in\Cdlg$,
$\Jump(\omega)\cap(\Bbb R\times[0,t])$ is countably infinite and
can be enumerated as $\sequencen{(\xi_{nt}(\omega),s_{nt}(\omega))}$ where
$|\xi_{n+1,t}(\omega)|<|\xi_{nt}(\omega)|$ for every $n$, for every $t>0$.
For any particular $t>0$, $\omega(t)=\sum_{n=0}^{\infty}\xi_{nt}(\omega)$ for
$\ddot\mu$-almost every $\omega$.

}{\leader{652N}{Alternative description of the Cauchy process (a)}
Let $\mu_0$ be the indefinite-integral
measure over $\mu_L$ corresponding to the function
$\xi\mapsto\Bover1{\pi\xi^2}:\BbbR\setminus\{0\}\to\coint{0,\infty}$%
\cmmnt{ (234J)},
$\mu_1$ the subspace measure on $\coint{0,\infty}$ induced by $\mu_L$, and
$\mu=\mu_0\times\mu_1$ the c.l.d.\ product measure on
$S=\Bbb R\times\coint{0,\infty}$\cmmnt{ (251F)}.   \cmmnt{We see that

%\query isn't \mu an indefinite-integral measure over 2-dim Leb m?

\inset{$\mu$ is atomless\cmmnt{ (234Nf, 251Xt)},

$\mu(\{\xi\}\times\coint{0,\infty})=\mu(\Bbb R\times\{t\})=0$ whenever
$\xi\in\Bbb R$ and $t\ge 0$,

$\mu(\{\xi:|\xi|\ge\delta\}\times[0,t])=\Bover{2t}{\pi\delta}$ is finite whenever
$\delta>0$ and $t\in\coint{0,\infty}$,

$\mu([0,\delta]\times[\alpha,\beta])=\mu([-\delta,0]\times[\alpha,\beta])
=\infty$ whenever $\delta>0$ and $0\le\alpha<\beta$.}
}%end of comment

\spheader 652Nb
Let $\nu$ be the Poisson point process on $(S,\mu)$ with intensity
$1$\cmmnt{ (495E\footnote{In earlier editions %to 2013
I used the word `density'.}),
so that $\nu$ is a probability measure on $\Omega=\Cal PS$}, and
$\Sigma$ its domain.   Then for $\nu$-almost every $\varpi\subseteq S$,

\Centerline{$\xi\ne 0$, $|\xi|\ne|\xi'|$ and $s\ne s'$
whenever $(\xi,s),\,(\xi',s')\in\varpi$ are distinct\dvro{,}{}}

\cmmnt{\noindent(applying 495H\Latereditions\ to $(\xi,s)\mapsto|\xi|$ and
$(\xi,s)\mapsto s$),}

\Centerline{$\varpi\cap\{(\xi,s):|\xi|\ge\delta$, $s\le t\}$
is finite whenever $\delta>0$ and $t\in\coint{0,\infty}$ are rational,}

$$\eqalign{&\varpi\cap([0,\delta]\times[\alpha,\beta])\text{ and }
\varpi\cap([-\delta,0]\times[\alpha,\beta])\text{ are infinite}\cr&
\mskip100mu\text{whenever }
\delta>0\text{ and }0\le\alpha<\beta\text{ and }\alpha,\,\beta\text{ and }
\delta\text{ are rational}.\cr}$$

\noindent Write $\Omega_0$ for the set of $\varpi\subseteq S$ with these
properties.   Note that if $\varpi\in\Omega_0$ then in fact

\Centerline{$\varpi\cap\{(\xi,s):|\xi|\ge\delta$, $s\le t\}$
is finite whenever $\delta>0$ and $t\in\coint{0,\infty}$,}

$$\eqalign{&\varpi\cap([0,\delta]\times[\alpha,\beta])\text{ and }
\varpi\cap([-\delta,0]\times[\alpha,\beta])\text{ are infinite}\cr&
\mskip100mu\text{whenever }
\delta>0\text{ and }0\le\alpha<\beta.\cr}$$

\spheader 652Nc For $t\ge 0$ set

\Centerline{$\Sigma_t=\{H\symmdiff A:H\in\Sigma$,
$H=\{\varpi:\varpi\subseteq S$, $\varpi\cap(\Bbb R\times[0,t])\in H\}$,
$A\in\Cal N(\mu)\}$.}

\noindent Then $\langle\Sigma_t\rangle_{t\ge 0}$ is a filtration of
$\sigma$-subalgebras of $\Sigma$.   Moreover, it is right-continuous.
\prooflet{\Prf\ Take any $t\ge 0$, and set $s_n=t+2^{-n}$ for $n\in\Bbb N$.
Setting
$S_0=\Bbb R\times[0,t]$, $S_1=\Bbb R\times\ooint{t+1,\infty}$ and
$S_n=\Bbb R\times\ocint{s_{n-2},s_{n-1}}$ for $n\ge 2$, we have an
isomorphism $\psi:\Cal PS\to\prod_{n\in\Bbb N}\Cal PS_n$ where
$\psi(\varpi)=\sequencen{\varpi\cap S_n}$ for $\varpi\subseteq S$, and
$\psi$ is a measure space isomorphism between $\nu$ and the product measure
$\tilde\lambda=\prod_{n\in\Bbb N}\nu_n$, where each $\nu_n$ is the Poisson point
process with intensity 1 on $S_n$ endowed with
the subspace measure induced by $\mu$
(495F).   Suppose that $H\in\bigcap_{s>t}\Sigma_s$ and
consider $\psi[H]$.   For each $n\in\Bbb N$, there is an $H_n\in\Sigma$ such
that
$H_n=\{\varpi:\varpi\subseteq S$, $\varpi\cap(\Bbb R\times[0,s_n])\in H_n\}$
and $H\symmdiff H_n$ is $\nu$-negligible.   Setting $H'_n=\bigcap_{m\ge n}H_m$,
we see that
$H'_n=\{\varpi:\varpi\subseteq S$, $\varpi\cap(\Bbb R\times[0,s_n])\in H'_n\}$
and $H\symmdiff H'_n$ is negligible for each $n$.   Now each
$\psi[H'_n]\subseteq\prod_{n\in\Bbb N}\Cal PS_n$ is determined by coordinates in
$\{0,n+2,n+3,\ldots\}$, so
$\bigcap_{m\in\Bbb N}\psi[H'_m]=\bigcap_{m\ge n}\psi[H'_m]$
is determined by coordinates in $\{0,n+2,n+3,\ldots\}$ for each $n$, and there
is therefore a $W\subseteq\prod_{n\in\Bbb N}\Cal PS_n$ determined by
the single coordinate $\{0\}$ such that
$W\symmdiff\bigcap_{m\in\Bbb N}\psi[H'_m]$ is $\tilde\lambda$-negligible, by
254Rd.  But now $H\symmdiff\psi^{-1}[W]$ is $\nu$-negligible, while
$\psi^{-1}[W]=\{\varpi:\varpi\subseteq S$,
$\varpi\cap(\Bbb R\times[0,t])\in\psi^{-1}[W]\}$.   So $H\in\Sigma_t$.
As $t$ and $H$ are arbitrary, $\langle\Sigma_t\rangle_{t\ge 0}$ is
right-continuous.\ \Qed}

\spheader 652N{d} Suppose that we have an $m\ge 0$ and family
$\langle E_{\alpha}\rangle_{0\le\alpha<1}$ of Borel subsets of
$[-1,1]\times[0,m]$ such that

\Centerline{$E_{\alpha}\subseteq E_{\beta}$ whenever $0\le\alpha\le\beta<1$,}

\Centerline{if $0\le\alpha<1$ and $(\xi,t)\in E_{\alpha}$ then
$(-\xi,t)\in E_{\alpha}$,}

\Centerline{if $0\le\alpha<1$ there is a $\delta>0$ such that
$|\xi|\ge\delta$ whenever $(\xi,t)\in E_{\alpha}$.}

\noindent For $0\le\alpha<1$ set

$$\eqalignno{X_{\alpha}(\varpi)&
=\sum_{(\xi,s)\in\varpi\cap E_{\alpha}}\xi
   \text{\quad if }\varpi\in\Omega_0,
\Displaycause{this is legitimate because $\varpi\cap E_{\alpha}$ is finite}
=0\text{\quad if }\varpi\in\Omega\setminus\Omega_0.\cr}$$

\noindent Then

(i) $\lim_{\alpha\uparrow 1}X_{\alpha}(\varpi)$ is defined in
$\Bbb R$ for almost every $\varpi$,

(ii) setting $h(\varpi)=\sup_{\alpha<1}|X_{\alpha}(\varpi)|$ for
$\varpi\in\Omega$, $(\int h\,d\nu)^2$ is defined and is not greater than
$\sup_{\alpha<1}\int_{E_{\alpha}}\xi^2\mu(d(\xi,s))$.

\leaveitout{\Centerline{$\int X_{\alpha}d\nu=0$,\quad
$\int X_{\alpha}^2d\nu=\int_{E_{\alpha}}\xi^2\mu(d(\xi,s))\le\bover{2\pi}m$}

\noindent for every
$\alpha\in\coint{0,1}$, $\langle X_{\alpha}\rangle_{0\le\alpha<1}$ is a
uniformly
integrable martingale, $X(\varpi)=\lim_{\alpha\uparrow 1}X_{\alpha}(\varpi)$
is defined in $\Bbb R$ for almost every $\varpi$, $X$ is
$\Sigma$-measurable and
$X^{\ssbullet}=\lim_{\alpha\uparrow 1}X_{\alpha}^{\ssbullet}$ in $L^0(\nu)$.}

\Prf{\bf (i)}\grheada\ If

\Centerline{$\Tau_{\alpha}=\{H:H\in\Sigma$,
  $H=\{\varpi:\varpi\in\Omega$, $\varpi\cap E_{\alpha}\in H\}\}$,}

\Centerline{
$\hat\Tau_{\alpha}=\{H\symmdiff A:H\in\Tau_{\alpha}$, $A\in\Cal N(\nu)\}$,}

\Centerline{$\Tau'_{\alpha}=\{H:H\in\Sigma$,
  $H=\{\varpi:\varpi\setminus E_{\alpha}\in H\}\}$,}

\Centerline{
$\hat\Tau{}'_{\alpha}=\{H\symmdiff A:H\in\Tau'_{\alpha}$, $A\in\Cal N(\nu)\}$,}

\noindent then $X_{\alpha}$ is $\hat\Tau_{\alpha}$-measurable.   We have

\Centerline{$\int_{E_{\alpha}}|\xi|\mu(d(\xi,s))
\le\mu E_{\alpha}<\infty$,}

$$\eqalignno{\int X_{\alpha}d\nu&
=\int_{E_{\alpha}}\xi\,\mu(d(\xi,s))
\Displaycause{495Oa\formerly{4{}95L}}
=\int_{E_{\alpha}}-\xi\,\mu(d(\xi,s))
\Displaycause{because $(\xi,t)\mapsto(-\xi,t):E_{\alpha}\to R_{\alpha}$
is a $\mu$-preserving bijection}
=0.\cr}$$

\noindent If $0<\beta<\alpha$, then

\Centerline{$X_{\beta}(\varpi)-X_{\alpha}(\varpi)
=\sum_{(\xi,s)\in\varpi\cap E_{\beta}\setminus E_{\alpha}}\xi$}

\noindent for $\varpi\in\Omega_0$.   So $X_{\beta}-X_{\alpha}$
is $\hat\Tau{}'_{\alpha}$-measurable
By 495F\Latereditions, $\Tau'_{\alpha}$ and $\Tau_{\alpha}$ are
independent, so $\hat\Tau{}'_{\alpha}$ and $\hat\Tau_{\alpha}$ are independent.
Since $\int X_{\beta}-X_{\alpha}d\nu=0$,
$\int_{H}X_{\beta}-X_{\alpha}d\nu=0$ for every $H\in\hat\Tau_{\alpha}$, so
$X_{\alpha}$ is a conditional expectation of $X_{\beta}$ on $\hat\Tau_{\alpha}$.
But this means that $\langle X_{\alpha}\rangle_{\alpha<1}$ is a martingale
with respect to the ordering $\le$ on $\coint{0,1}$.

\medskip

\qquad\grheadb\ Next, 495Ob\Latereditions\ tells us that

$$\eqalign{\int X^2_{\alpha}d\nu&
=\int_{E_{\alpha}}\xi^2\mu(d(\xi,s))
    +(\int_{E_{\alpha}}\xi\,\mu(d(\xi,s)))^2\cr&
=\int_{E_{\alpha}}\xi^2\mu(d(\xi,s))
\le m\int_{[-1,1]}\xi^2\mu_0(d\xi)
=\Bover{2m}{\pi}\cr}$$

\noindent for every $\alpha\in\coint{0,1}$.   So $\{X_{\alpha}:0\le\alpha<1\}$
is $\|\,\|_2$-bounded, therefore uniformly integrable (621Be).
By 622J, $v=\lim_{\alpha\uparrow 1}X_{\alpha}^{\ssbullet}$ is defined
in $L^0(\nu)$ for the
topology of convergence in measure and
$\inf_{\alpha<1}\sup_{\alpha\le\beta<1}|v-X_{\beta}^{\ssbullet}|=0$.

\medskip

\qquad\grheadc\ If we take any $\Sigma$-measurable function
$g:\Omega\to\Bbb R$ such that $g^{\ssbullet}=v$ and write $Q$ for
$\Bbb Q\cap\coint{0,1}$, we know that

\Centerline{$\inf_{q\in Q}\sup_{q'\in Q,q'\ge q}
   |g^{\ssbullet}-X_{q'}^{\ssbullet}|=0$
in $L^0(\frak A)$}

\noindent and because $Q$ is countable, this translates directly to

\Centerline{$\inf_{q\in Q}\sup_{q'\in Q,q'\ge q}|g(\varpi)-X_{q'}(\varpi)|=0$
for $\nu$-almost every $\varpi\subseteq\Cal S$}

\noindent if we calculate the suprema and infimum in $[0,\infty]$.   But now
observe that, at least for $\varpi\in\Omega_0$,

\Centerline{$\sup_{q'\in Q,q'\ge q}|h(\varpi)-X_{tq'}(\varpi)|
=\sup_{q\le\beta<1}|g(\varpi)-X_{\beta}(\varpi)|$}

\noindent for every $q\in Q$, because if $q\le\beta<1$ and
$\varpi\in\Omega_0$, there is a $q'\in Q$ such that $\beta<q'<$1 and
$\varpi\cap(E_{q'}\setminus E_{\beta})=\emptyset$ so that
$X_{\beta}(\varpi)=X_{q'}(\varpi)$.   And now we see that for
$\nu$-almost every $\varpi\in\Omega_0$,

\Centerline{$\inf_{\alpha<1}
\sup_{\alpha\le\beta<1}|g(\varpi)-X_{\beta}(\varpi)|=0$}

\noindent and $g(\varpi)=\lim_{\alpha\uparrow 1}X_{\alpha}(\varpi)$.

\leaveitout{Set

\Centerline{$\Omega_1=\{\varpi:\varpi\in\Omega_0$,
$\lim_{\alpha\uparrow 1}X_{\alpha}(\varpi)$ is defined in $\Bbb R\}$}

\noindent and

\Centerline{$X(\varpi)=\lim_{\alpha\uparrow 1}X_{\alpha}(\varpi)$}

\noindent for $\varpi\in\Omega_1$.   Then $\Omega_1$ is conegligible and
$X$ is $\Sigma$-measurable, while
$X^{\ssbullet}=\lim_{\alpha\uparrow 1}X_{\alpha}^{\ssbullet}$.}

\medskip

\quad{\bf (ii)} As noted in (i-$\gamma$) just above, if $\varpi\in\Omega_0$ and
$0\le\alpha<1$ there is a rational $q\in\coint{\alpha,1}$ such that
$X_{\alpha}(\varpi)=X_q(\varpi)$, so
$h(\varpi)=\sup_{q\in\Bbb Q,0\le q<1}|X_q(\varpi)|$ for every
$\varpi\in\Omega_0$.   Now 623M tells us that
$\sup_{0\le\alpha<1}|X_{\alpha}^{\ssbullet}|$ is defined
in $L^0(\nu)$ and

$$\eqalignno{(\int h\,d\nu)^2&
=\|\sup_{q\in\Bbb Q,0\le q<1}|X_q^{\ssbullet}|\|_1^2
\le\|\sup_{0\le\alpha<1}|X_{\alpha}^{\ssbullet}|\|_1^2\cr&
\le\sup_{0\le\alpha<1}\|X_{\alpha}^{\ssbullet}\|_2^2
=\sup_{\alpha<1}\int_{E_{\alpha}}\xi^2\mu(d(\xi,s))\cr}$$

\noindent by (i-$\beta$).\ \Qed

\medskip

\spheader 652N{e}{\bf (i)}
For $m$, $n\in\Bbb N$, $\alpha\in\coint{0,1}$ and $\varpi\in\Omega$ set

\Centerline{$E_{mn\alpha}=\{(\xi,s):2^{-n-1}<|\xi|\le 2^{-n}$,
$0\le s\le\alpha m\}\subseteq S$,}

$$\eqalign{Y_{mn\alpha}(\varpi)
&=\sum_{(\xi,s)\in\varpi\cap E_{mn\alpha}}\xi
\text{ if }\varpi\in\Omega_0,\cr
&=0\text{ otherwise.}\cr}$$

\noindent Then

$$\eqalignno{\int_{E_{mn\alpha}}\xi^2\mu(d(\xi,s))&
=\alpha m\int_{F_n}\xi^2\mu_0(d\xi)
\Displaycause{where $F_n=\{\eta:2^{-n-1}<|\eta|\le 2^{-n}\}$}
=\Bover{\alpha m}{\pi}\mu_LF_n
=\Bover{2^{-n}\alpha m}{\pi}.\cr}$$

If we set

\Centerline{$h_{mn}(\varpi)
=\sup_{\alpha\in\Bbb Q\cap\coint{0,1}}|Y_{mn\alpha}(\varpi)|\in[0,\infty]$}

\noindent for $\varpi\in\Omega$,
$\Expn(h_{mn})\le 2^{-n/2}\sqrt{\bover{m}{\pi}}$, by (d-ii).
But this means that

\Centerline{$\Omega_2
=\{\varpi:\varpi\in\Omega_0$, $\sum_{n=0}^{\infty}h_{mn}(\varpi)<\infty$
   for every $m\in\Bbb N\}$}

\noindent is $\nu$-conegligible.   At the same time, for $\varpi\in\Omega_2$,

\Centerline{$|Y_{mn\alpha}(\varpi)|
\le\sup_{q\in\Bbb Q\cap\coint{0,1}}|Y_{mnq}(\varpi)|
\le h_{mn}(\varpi)$}

\noindent whenever $m$, $n\in\Bbb N$ and $\alpha\in\coint{0,1}$, just as in
(d-i-$\gamma$) and (d-ii) above.

\medskip

\quad{\bf (ii)} For $t\ge 0$ and $n\in\Bbb N$, set

\Centerline{$E_{tn}=\{(\xi,s):2^{-n-1}<|\xi|\le 2^{-n}$,
$0\le s\le t\}\subseteq S$,}

$$\eqalign{Y_{tn}(\varpi)
&=\sum_{(\xi,s)\in\varpi\cap E_{tn}}\xi
\text{ if }\varpi\in\Omega_0,\cr
&=0\text{ otherwise.}\cr}$$

\noindent If $\varpi\in\Omega_2$ then,
applying (i) with $m=\lfloor t+1\rfloor$ and $\alpha=\bover{t}{m}$,
we see that $\sum_{n=0}^{\infty}Y_{tn}(\varpi)$ is defined
(as an unconditional sum) and finite.
Since $\varpi\cap(\ooint{1,\infty}\times[0,t])$
and $\varpi\cap(\ooint{-\infty,-1}\times[0,t])$ are finite,

\Centerline{$Z_t(\varpi)
=\sum_{n=0}^{\infty}Y_{tn}(\varpi)+\sum_{(\xi,s)\in\varpi,|\xi|>1,s\le t}\xi$}

\noindent is defined.   We see also that, for any $m\in\Bbb N$,

\Centerline{$Z_t(\varpi)
=\lim_{n\to\infty}\sum_{k=0}^nY_{tk}(\varpi)
     +\sum_{(\xi,s)\in\varpi,|\xi|>1,s\le t}\xi$}

\noindent uniformly for $t\in[0,m]$.   Since the functions

\Centerline{$t\mapsto\sum_{k=0}^nY_{tk}(\varpi)
     +\sum_{(\xi,s)\in\varpi,|\xi|>1,s\le t}\xi$}

\noindent are \cadlag\ for every $n\in\Bbb N$,
$t\mapsto Z_t(\varpi):[0,m]\to\Bbb R$ is \cadlag.   Since $m$ is arbitrary,
$t\mapsto Z_t(\varpi):\coint{0,\infty}\to\Bbb R$ is \cadlag.

For completeness, I will set $Z_t(\varpi)=0$ for $t\ge 0$ and
$\varpi\in\Omega\setminus\Omega_2$, so that $t\mapsto Z_t(\varpi)$ is \cadlag\
for every $\varpi\in\Omega$.

\spheader 652N{f} If $t\ge 0$, the characteristic function of $Z_t$ is
$\eta\mapsto e^{-t|\eta|}$.   \Prf\ For $n\in\Bbb N$, set

\Centerline{$F_{tn}=\{(\xi,s):2^{-n}\le|\xi|\le 2^n$, $s\le t\}
\subseteq S$,}

$$\eqalign{g_n(\xi,s)
&=\xi\text{ if }(\xi,s)\in F_{tn},\cr
&=0\text{ for other }(\xi,s)\in S,\cr}$$

$$\eqalign{Z_{tn}(\varpi)
&=\sum_{(\xi,s)\in\varpi\cap F_{tn}}\xi\text{\quad if }
   \varpi\cap F_{tn}\text{ is finite},\cr
&=0\text{ for other }\varpi\subseteq S.\cr}$$

\noindent Then $Z_t(\varpi)=\lim_{n\to\infty}Z_{tn}(\varpi)$ for every
$\varpi\in\Omega_2$.   Now the characteristic function of $Z_{tn}$ is

\Centerline{$\eta\mapsto\int_{\Cal PS}e^{i\eta Z_{tn}}d\nu
=\exp(\int_S(e^{i\eta g_n}-1)d\mu)$}

\noindent by 495P\formerly{4{}95M}.   Next,

$$\eqalign{\int_S(e^{i\eta g_n}-1)d\mu&
=\int_0^t\bigl(\int_{-2^n}^{-2^{-n}}
        \Bover1{\pi\xi^2}(e^{i\eta\xi}-1)d\xi
  +\int_{2^{-n}}^{2^n}
        \Bover1{\pi\xi^2}(e^{i\eta\xi}-1)d\xi\bigr)dt\cr&
=\Bover1{\pi}\int_0^t\int_{2^{-n}}^{2^n}
   \Bover1{\xi^2}(e^{i\eta\xi}+e^{-i\eta\xi}-2)d\xi dt\cr&
=\Bover{2t}{\pi}\int_{2^{-n}}^{2^n}
   \Bover1{\xi^2}(\cos\eta\xi-1)d\xi\cr&
=-\Bover{2t}{\pi}\int_{2^{-n}}^{2^n}
   \Bover1{\xi^2}(1-\cos|\eta|\xi)d\xi.\cr}$$

\noindent As in (b-ii) of the proof of 652M,
$\int_0^{\infty}\Bover1{\xi^2}(1-\cos|\eta|\xi)d\xi
=|\eta|\int_0^{\infty}\Bover1{\xi^2}(1-\cos\xi)d\xi$ is finite,
so we can apply Lebesgue's Dominated Convergence Theorem again to see that

$$\eqalignno{\lim_{n\to\infty}\int_S(e^{i\eta g_n}-1)d\mu&
=-\lim_{n\to\infty}\Bover{2t}{\pi}\int_{2^{-n}}^{2^n}
   \Bover1{\xi^2}(1-\cos|\eta|\xi)d\xi
=-\Bover{2t}{\pi}\int_0^{\infty}\Bover1{\xi^2}(1-\cos|\eta|\xi)d\xi\cr&
=-\Bover{2t|\eta|}{\pi}\int_0^{\infty}\Bover1{\xi^2}(1-\cos\xi)d\xi
=-\Bover{2t|\eta|}{\pi}\cdot\Bover{\pi}2
\Displaycause{652M(b-ii)}
=-t|\eta|,\cr}$$

\noindent and

$$\eqalignno{\int_{\Cal PS}e^{i\eta Z_t}d\nu&
=\lim_{n\to\infty}\int_{\Cal PS}e^{i\eta Z_{tn}}d\nu
=\lim_{n\to\infty}\exp(\int_S(e^{i\eta g_n}-1)d\mu)\cr&
=\exp(\lim_{n\to\infty}\int_S(e^{i\eta g_n}-1)d\mu)
=\exp(-t|\eta|),\cr}$$

\noindent as claimed.\ \Qed

By 285Mb, it follows that (for $t>0$) the distribution of $Z_t$ is the
Cauchy distribution $\lambda_t$ as defined in 652M.

\medskip

\spheader 652N{g}{\bf (i)} If $s$, $t>0$,
$Z_{s+t}-Z_s$ has the same distribution as $Z_t$.
\Prf\ Setting $\phi_s(\xi,t')=(\xi,s+t')$ for $\xi\in\Bbb R$ and $t'\ge 0$,
$\phi_s$ is a measure-preserving bijection
between $(S,\mu)$ and $\Bbb R\times\coint{s,\infty}$ with its subspace measure,
so $\varpi\mapsto\phi_s^{-1}[\varpi]:\Cal PS\to\Cal PS$ is $\nu$-\imp\
(495F), while (in the notation of (e) above)

\Centerline{$\sum_{(\xi,t')\in\varpi\cap F_{s+t,n}\setminus F_{sn}}\xi
=\sum_{(\xi,t')\in\phi_s^{-1}[\varpi]\cap F_{tn}\setminus F_{0n}}\xi$}

\noindent for every $\varpi\in\Omega_0$.   So $Z_{s+t,n}-Z_{sn}$ and
$Z_{tn}-Z_{t0}=Z_{tn}$ have the same distribution for each $n$.
Since $\sequencen{Z_{t'n}}$ converges in measure to $Z_{t'}$
for every $t'\ge 0$, 454U\Latereditions\ tells us that
$Z_{s+t}^{\ssbullet}-Z_s^{\ssbullet}$ and $Z_t^{\ssbullet}$ have the same
distribution, that is, $Z_{s+t}-Z_s$ and $Z_t$ have the same distribution.\ \Qed

\medskip

\quad{\bf (ii)} If $s$, $t\ge 0$ and we take $\Sigma'_s$ to be

\Centerline{$\{H\symmdiff A:H\in\Sigma$,
$H=\{\varpi:\varpi\subseteq S$,
   $\varpi\cap(\Bbb R\times\ooint{s,\infty})\in H\}$,
$A\in\Cal N(\mu)\}$,}

\noindent then $\Sigma'_s$ and $\Sigma_s$, as defined in (c), are independent,
by 495F again, while $Z_{s+t}-Z_s$ is $\Sigma'_s$-measurable.   So
$Z_{s+t}-Z_s$ is independent of $\Sigma_s$.

\medskip

\spheader 652Nh Accordingly
we have a second representation of the Cauchy process
based on $(\Omega,\nu)$ rather than on $(\Cdlg,\ddot\mu)$.
We saw in (e-ii) that $t\mapsto Z_t(\varpi)$ is \cadlag\ for every
$\varpi\in\Omega$.   Taking $\frak A_t$ to be
$\{E^{\ssbullet}:E\in\Sigma_t\}$ for $t\ge 0$,
631D tells us that in the stochastic integration structure
$(\frak A,\bar\nu,\coint{0,\infty},\langle\frak A_t\rangle_{t\ge 0})$
we have a locally near-simple
process $\pmb z=\family{\sigma}{\Cal T_f}{z_{\sigma}}$ such that
$z_{\check t}=Z_t^{\ssbullet}$ for every $t\ge 0$.
Because $\langle\Sigma_t\rangle_{t\ge 0}$ is right continuous ((c) above),
so is $\langle\frak A_t\rangle_{t\ge 0}$.
Thus $\pmb z$ is a L\'evy process as defined in 652B.

\medskip

\spheader 652Ni We can use this new representation to get some information
about typical sample paths in the process described in 652Mc.

If $\varpi\in\Omega_0$ and $t>0$
we have a canonical enumeration
$\sequencen{(\tilde\xi_{nt}(\varpi),\tilde s_{nt}(\varpi))}$ of the countably
infinite set $\varpi\cap(\Bbb R\times[0,t])$ such
that $|\tilde\xi_{n+1,t}(\varpi)|<|\tilde\xi_{nt}(\varpi)|$
for every $n\in\Bbb N$.
Now we find that for any $t\ge 0$, the conditional sum

\Centerline{$\sum_{n=0}^{\infty}\tilde\xi_{nt}(\varpi)
=\lim_{n\to\infty}\sum_{i=0}^n\tilde\xi_{it}(\varpi)$}

\noindent is defined in $\Bbb R$ and equal to $Z_t(\varpi)$
for almost every $\varpi$.
\prooflet{\Prf\ For $0\le\alpha<1$, set
$E_{\alpha}=([-1,-1+\alpha]\cup[1-\alpha,1])\times[0,t]$.   By
(d-i),

\Centerline{$\Omega_{1t}=\{\varpi:\varpi\in\Omega_0$,
$\lim_{\alpha\uparrow 1}\sum_{(\xi,s)\in\varpi\cap E_{\alpha}}\xi$ is defined in
$\Bbb R\}$}

\noindent is $\nu$-conegligible.   Now if $\varpi\in\Omega_{1t}\cap\Omega_2$ and
$\epsilon>0$,
$\varpi\cap((\ooint{-\infty,-1}\cup\ooint{1,\infty})\times[0,t]$ is finite,
so there is a first
$m\in\Bbb N$ such that $|\tilde\xi_{mt}|(\varpi)\le 1$,   Next, there is a
$\beta\in\coint{0,1}$ such that

\Centerline{$\epsilon\ge|\sum_{(\xi,s)\in\varpi\cap E_{\alpha'}}\xi
   -\sum_{(\xi,s)\in\varpi\cap E_{\alpha}}\xi|
=|\sum_{(\xi,s)\in\varpi\cap(E_{\alpha'}\setminus E_{\alpha})}\xi|$}

\noindent whenever $\beta\le\alpha\le\alpha'<1$.   Taking
$k\ge m$ such that $|\tilde\xi_{kt}(\varpi)|\le\beta$, we see that
if $k\le n\le n'$ then

\Centerline{$|\sum_{i=0}^{n'}\tilde\xi_{it}(\varpi)
  -\sum_{i=0}^n\tilde\xi_{it}(\varpi)|
=|\sum_{(\xi,s)\in\varpi\cap(E_{\alpha'}\setminus E_{\alpha})}\xi|\le\epsilon$}

\noindent where $\alpha=1-|\tilde\xi_{nt}|$ and $\alpha'=1-|\tilde\xi_{n't}|$.
As $\epsilon$ is arbitrary,
$\lim_{n\to\infty}\sum_{i=0}^n\tilde\xi_{it}(\varpi)$ is defined.

At the same time,

$$\eqalign{Z_t(\varpi)&
=\lim_{n\to\infty}\sum_{\Atop{(\xi,s)\in\varpi,s\le t}{|\xi|>2^{-n}}}\xi
=\sum_{(\xi,s)\in\varpi,\xi>1}\xi
   +\lim_{n\to\infty}
      \sum_{\Atop{(\xi,s)\in\varpi,s\le t}{2^{-n}<|\xi|\le 1}}\xi\cr&
=\sum_{(\xi,s)\in\varpi,\xi>1}\xi
   +\lim_{\alpha\uparrow 1}
      \sum_{\Atop{(\xi,s)\in\varpi,s\le t}{1-\alpha<|\xi|\le 1}}\xi
=\lim_{n\to\infty}\sum_{i=0}^n\tilde\xi_{it}(\varpi).  \text{ \Qed}\cr}$$
}

\spheader 652N{j} By (e-ii),
we have a function $\phi:\Omega\to\Cdlg$ defined by saying that

$$\eqalign{\phi(\varpi)(t)
&=Z_t(\varpi)\text{ if }\varpi\in\Omega_2,\cr
&=0\text{ otherwise}.\cr}$$

\noindent Now $\phi$ is \imp\ for $\nu$ and the measure $\ddot\mu$ defined in
652Mc.   \Prf\ If $0=s_0<\ldots<s_n$ in $\coint{0,\infty}$ and
$\alpha_i\in\Bbb R$ for $i\le n$, then

$$\eqalignno{\nu\phi^{-1}\{\omega&:\omega\in\Cdlg,\,\omega(s_0)\le\alpha_0,\,
   \omega(s_i)-\omega(s_{i-1})\le\alpha_i\text{ for }1\le i\le n\}\cr&
=\nu\{\varpi:\varpi\in\Omega_2,\,Z_{s_0}(\varpi)\le\alpha_0,\,
   Z_{s_i}(\varpi)-Z_{s_{i-1}}(\varpi)\le\alpha_i\text{ for }1\le i\le n\}\cr&
=0\text{ if }\alpha_0<0,\cr&
=\prod_{1\le i\le n}
     \nu\{\varpi:Z_{s_i}(\varpi)-Z_{s_{i-1}}(\varpi)\le\alpha_i\}
\Displaycause{by (g-ii)}
\mskip100mu=\prod_{1\le i\le n}
     \nu\{\varpi:Z_{s_i-s_{i-1}}(\varpi)\le\alpha_i\}
\Displaycause{by (g-i)}
\mskip100mu=\prod_{1\le i\le n}
     \lambda_{s_i-s_{i-1}}\ocint{-\infty,\alpha_i}\text{ otherwise}\cr}$$

\noindent (by (f)), and in either case is equal to

\Centerline{$\ddot\mu\{\omega:\omega(s_0)\le\alpha_0,\,
   \omega(s_i)-\omega(s_{i-1})\le\alpha_i\text{ for }1\le i\le n\}$.}

\noindent Since
$\Cal H=\{H:H\in\ddot\Sigma$, $\nu\phi^{-1}[H]$ is defined and equal to
$\ddot\mu H\}$ is a Dynkin class, it contains all sets of the form
$\{\omega:\omega\in\Cdlg,\,\omega(s_0)\in E_0,\,
   \omega(s_i)-\omega(s_{i-1})\in E_i\text{ for }1\le i\le n\}$ where
$n\in\Bbb N$ and $E_i\subseteq\Bbb R$ is Borel for every $i\le n$, and therefore
the $\sigma$-algebra generated by these, which contains all sets of the form
$\{\omega:\omega\in\Cdlg,\,\omega(s_i)\in E_i\text{ for }i\le n\}$ where
$n\in\Bbb N$ and $E_i\subseteq\Bbb R$ is Borel for every $i\le n$, and therefore
is the subspace $\sigma$-algebra defined by the Baire $\sigma$-algebra of
$\BbbR^{\coint{0,\infty}}$.   Since $\ddot\mu$ is a completion regular
quasi-Radon probability measure, every $H\in\ddot\Sigma$ is expressible as
$H_0\symmdiff A$ where $H_0\in\Cal H$ and $A$ is included in a
$\ddot\mu$-negligible member of $\Cal H$;  because $\nu$ is complete,
$\ddot\Sigma\subseteq\Cal H$, that is, $\phi$ is \imp.\ \Qed

\spheader 652N{k} If $\varpi\in\Omega_2$ and $t\ge 0$, then

$$\eqalign{\phi(\varpi)(t)-\lim_{s\uparrow t}\phi(\varpi)(s)
&=\xi\text{ if }(\xi,t)\in\varpi,\cr
&=0\text{ if there is no }\xi\text{ such that }(\xi,t)\in\varpi.\cr}$$

\noindent\Prf\ Take $m\in\Bbb N$ such that $t\le m$, and $\epsilon>0$.
Let $k\in\Bbb N$ be such that $\sum_{n=k}^{\infty}h_{mn}(\varpi)\le\epsilon$,
where $h_{mn}$ is defined as in (e-i) above.   For $s\ge 0$ set
$f(s)=\sum_{(\xi,s')\in\varpi,|\xi|>2^{-k},s'\le s}\xi$;  as
$\varpi\in\Omega_0$, this is a finite sum.   Then

\Centerline{$|\phi(\varpi)(s)-f(s)|=|Z_s(\varpi)-f(s)|\le\epsilon$}

\noindent for every $s\le m$, while

$$\eqalign{f(t)-\lim_{s\uparrow t}f(s)
&=\xi\text{ if }(\xi,t)\in\varpi,\cr
&=0\text{ if there is no }\xi\text{ such that }(\xi,t)\in\varpi.\cr}$$

\noindent Now

$$\eqalign{&\limsup_{s\uparrow t}
    |\phi(\varpi)(t)-\xi-\phi(\varpi)(s)|\cr&
\mskip70mu\le|\phi(\varpi)(t)-f(t)|
    +\limsup_{s\uparrow t}|f(t)-f(s)-\xi|
    +\limsup_{s\uparrow t}|f(s)-\phi(\varpi)(s)|\cr&
\mskip70mu\le 2\sup_{s\le t}|\phi(\varpi)(s)-f(s)|
\le 2\epsilon\cr}$$

\noindent if $(\xi,t)\in\varpi$, and

\Centerline{$\limsup_{s\uparrow t}|\phi(\varpi)(t)-\phi(\varpi)(s)|
\le 2\epsilon$}

\noindent if there is no $\xi$ such that $(\xi,t)\in\varpi$.   As $\epsilon$ is
arbitrary, we have the result.\ \Qed

\spheader 652N{l} If $\omega\in\Cdlg$, I will write $\Jump(\omega)$ for

\Centerline{$\{(\xi,t):\xi\in\Bbb R\setminus\{0\}$, $t>0$,
$\omega(t)=\lim_{s\uparrow t}\omega(s)+\xi\}$.}

\noindent Observe that $\varpi=\Jump(\phi(\varpi))$ for every
$\varpi\in\Omega_2$, by (k).

For $E\subseteq S$, set

\Centerline{$A_1(E)=\{\omega:E\cap\Jump(\omega)\ne\emptyset\}$.}

\noindent If $E\in\Cal B(S)$, $A_1(E)\in\ddot\Sigma$.
\Prf\ Since $(\omega,t)\mapsto\omega(t):\Cdlg\times\coint{0,\infty}\to\Bbb R$ is
$\ddot\Sigma\tensorhat\Cal B(\coint{0,\infty})$-measurable (631D),

\Centerline{$\{(\omega,\xi,t,s):|\omega(t)-\omega(s)-\xi|\le 2^{-n}\}$}

\noindent belongs to
$\ddot\Sigma\tensorhat\Cal B(S\times\coint{0,\infty})$ for every $n\in\Bbb N$,

$$\eqalign{\{(\omega,\xi,t,\sequencen{s_n}):
(\xi,t)&\in E,\,t-2^{-n}\le s_n<t\cr&
\text{ and }
|\omega(t)-\omega(s_n)-\xi|\le 2^{-n}\text{ for every }n\in\Bbb N\}\cr}$$

\noindent belongs to

\Centerline{$\ddot\Sigma\tensorhat\Cal B(S\times\coint{0,\infty}^{\Bbb N})$}

\noindent and its projection $A_1(E)$
can be obtained by Souslin's operation from
members of $\ddot\Sigma$ (423O),
so belongs to $\ddot\Sigma$ (431A again).\ \QeD\   Now (k) tells us that

$$\eqalign{\ddot\mu A_1(E)
&=\nu\phi^{-1}[A_1(E)]=\nu(\Omega_2\cap\phi^{-1}[A_1(E)])\cr&
=\nu\{\varpi:\varpi\in\Omega_2,\,E\cap\varpi\ne\emptyset\}\cr&
\mskip50mu=1-e^{-\mu E}\text{ if }\mu E\text{ is finite},\cr&
\mskip50mu=1\text{ if }\mu E=\infty.\cr}$$

\spheader 652N{m} Observe now that $\mu$ is atomless and
$(S,\mu)$ is a countably separated measure space
(343D), while $\ddot\mu$ is complete.   So
$\Jump:\Cdlg\to\Cal PS$ leads us to a complete
image measure $\ddot\mu\Jump^{-1}$ which agrees with $\nu$ on sets of the form
$\{S:S\cap E\ne\emptyset\}$ where $\mu E<\infty$, and therefore extends $\nu$
(495I\formerly{4{}95Xa});  that is, $\Jump$ is \imp\ for $\ddot\mu$ and $\nu$.
We can therefore refer to (b) above to see that, for $\ddot\mu$-almost every
$\omega\in\Cdlg$,

\Centerline{$\xi\ne 0$, $|\xi|\ne|\xi'|$ and $s\ne s'$
whenever $(\xi,s),\,(\xi',s')\in\Jump(\omega)$ are distinct,}

\Centerline{$\Jump(\omega)\cap\{(\xi,s):|\xi|\ge\delta$, $s\le t\}$
is finite whenever $\delta>0$ and $t\in\coint{0,\infty}$,}

$$\eqalign{&\Jump(\omega)\cap([0,\delta]\times[\alpha,\beta])\text{ and }
\varpi\cap([-\delta,0]\times[\alpha,\beta])\text{ are infinite}\cr&
\mskip100mu\text{whenever }
\delta>0\text{ and }0\le\alpha<\beta,\cr}$$

\noindent so that for $\ddot\mu$-almost every $\omega$
we have, for every $t>0$, a canonical enumeration
$\sequencen{(\xi_{nt}(\omega),s_{nt}(\omega))}$ of the countably
infinite set $\Jump(\omega)\cap(\Bbb R\times[0,t])$ such
that $|\xi_{n+1,t}(\omega)|<|\xi_{nt}(\omega)|$ for every $n\in\Bbb N$;
we take $\xi_{nt}(\omega)=\tilde\xi_{nt}(\Jump(\omega))$,
$s_{nt}(\omega)=\tilde s_{nt}(\Jump(\omega))$ whenever
$\Jump(\omega)\in\Omega_0$ and $n\in\Bbb N$.

If $\varpi\in\Omega_2$, then $\varpi=\Jump(\phi(\varpi))$, so
$\xi_{nt}(\phi(\varpi))=\tilde\xi_{nt}(\varpi)$
and $s_{nt}(\phi(\varpi))=\tilde s_{nt}(\varpi)$ for all $t>0$ and $n\in\Bbb N$.
For any particular $t>0$, $\Omega_2\cap\Omega_{1t}$ is conegligible
and

\Centerline{$\sum_{n=0}^{\infty}\xi_{nt}(\phi(\varpi))
=\sum_{n=0}^{\infty}\tilde\xi_{nt}(\varpi)
=Z_t(\varpi)=\phi(\varpi)(t)$.}

\noindent So $\omega(t)=\sum_{n=0}^{\infty}\xi_{nt}(\omega)$ whenever
$\omega\in\phi[\Omega_2\cap\Omega_{1t}]$.   On the other hand,
$\omega\mapsto\xi_{nt}(\omega)=\tilde\xi_{nt}(\Jump(\omega))$ is
$\ddot\Sigma$-measurable for each $n$ because $\Jump$ is
$(\ddot\Sigma,\Sigma)$-measurable and $\tilde\xi_{nt}$ is $\Sigma$-measurable.
But this implies that
$\{\omega:\omega(t)=\sum_{n=0}^{\infty}\xi_{nt}(\omega)\}$ belongs to
$\ddot\Sigma$;  as it includes the set $\phi[\Omega_2\cap\Omega_{1t}]$
of full outer measure, it is conegligible.   So we see that
$\omega(t)=\sum_{n=0}^{\infty}\xi_{nt}(\omega)$ for $\ddot\mu$-almost every
$\omega\in\Cdlg$.
}%end of dvro

\dvro{\leader{652O}{Third construction for the Cauchy process (a)}
Write $\mu_W=\mu_{W2}$ for two-dimensional Wiener measure on the space
$\Omega=C(\coint{0,\infty};\BbbR^2)_0$
of continuous functions from $\coint{0,\infty}$ to $\BbbR^2$ starting at zero.
%endowed with the Polish topology of uniform
%convergence on compact subsets of $\coint{0,\infty}$.
For $\omega\in\Omega$, write $\omega_0$,
$\omega_1$ for its coordinates in $C(\coint{0,\infty})_0$.

\spheader 652Ob For $t\ge 0$, let $\tilde h_t$ be the Brownian hitting time to
$\ooint{t,\infty}\times\Bbb R$, and set

$$\eqalign{\tilde Z_t(\omega)
&=\omega_1(\tilde h_t(\omega))\text{ if }\omega_0[\coint{0,\infty}\,]=\Bbb R,\cr
&=0\text{ otherwise}.\cr}$$

\noindent Then $\langle\tilde Z_t\rangle_{t\ge 0}$ has the same distribution as
the process $\langle Z_t\rangle_{t\ge 0}$ of 652N, and
$t\mapsto\tilde Z_t(\omega)$ is \cadlag\ for every $\omega\in\Omega$.

\spheader 652Oc Writing $\Sigma$ for the domain of $\mu_W$,

\Centerline{$\Sigma_t=\{E:E\in\Sigma$, $\omega'\in E$ whenever $\omega\in E$,
$\omega'\in\Omega$ and $\omega'\restr[0,t]=\omega\restr[0,t]\}$,}

\Centerline{$\Sigma_{\tilde h_t}
=\{E:E\in\Omega$,
$E\cap\{\omega:\tilde h_t(\omega)\le s\}\in\Sigma_s$ for every
$s\ge 0\}$,}

\Centerline{$\Tau_t=\{E\symmdiff A:E\in\bigcap_{s>t}\Sigma_{\tilde h_s}$,
$A\in\Cal N(\mu_W)\}$}

\noindent for $t\ge 0$, $\langle\Tau_t\rangle_{t\ge 0}$ is a right-continuous
filtration of $\sigma$-algebras.   $\tilde Z_t$ is $\Tau_t$-measurable and
$\tilde Z_s-\tilde Z_t$ is independent of $\Tau_t$ whenever $0\le t\le s$.

\spheader 652Od Applying the construction of 631D to $(\Omega,\Sigma,\mu_W)$,
$\langle\Tau_t\rangle_{t\ge 0}$ and $\langle\tilde Z_t\rangle_{t\ge 0}$ we get a
L\'evy process $\langle\tilde z_{\sigma}\rangle_{\sigma\in\Cal T_f}$ in which
the distribution of $\tilde z_{\check t}$ is the Cauchy distribution with scale
parameter $t$ for every $t\ge 0$.

}
{\leader{652O}{Third construction for the Cauchy process (a)(i)}
Write $\mu_W=\mu_{W2}$ for two-dimensional Wiener measure on the space
$\Omega=C(\coint{0,\infty};\BbbR^2)_0$
of continuous functions from $\coint{0,\infty}$ to $\BbbR^2$ starting at zero.
%endowed with the Polish topology of uniform
%convergence on compact subsets of $\coint{0,\infty}$ (477D).
Recall that $\Omega$ can be identified with $C(\coint{0,\infty})_0^2$ and
$\mu_W$ with $\mu_{W1}^2$, where $\mu_{W1}$ is one-dimensional Wiener measure
on $C(\coint{0,\infty})_0$.   For $\omega\in\Omega$, I will write $\omega_0$,
$\omega_1$ for its coordinates in $C(\coint{0,\infty})_0$, so that
$\omega(t)=(\omega_0(t),\omega_1(t))$ for $t\ge 0$.
Write $\Sigma$ for the domain of $\mu_W$.
For $t\ge 0$, let $\Sigma_t$ be
the $\sigma$-algebra of sets $F\in\Sigma$
such that $\omega'\in F$ whenever $\omega\in F$, $\omega'\in\Omega$ and
$\omega'\restr[0,t]=\omega\restr[0,t]$.

\medskip

\quad{\bf (ii)} For $t\ge 0$, let
$h_t$ be the Brownian hitting time to the closed set
$\coint{t,\infty}\times\Bbb R$ (477I), so that
$h_t(\omega)=\inf\{s:\omega_0(s)\ge t\}$, counting $\inf\emptyset$ as
$\infty$, and $\omega_0(h_t(\omega))=t$ if
$h_t(\omega)$ is finite.
Evidently $t\mapsto h_t(\omega)$ is non-decreasing.   Set

\Centerline{$\Omega_0'=\{\omega:\omega\in C([\coint{0,\infty})_0$,
 $\omega_0[\coint{0,\infty}\,]=\Bbb R\}$,
\quad$\Omega'=\Omega_0'\times C([\coint{0,\infty})_0$;}

\noindent then $\Omega_0'$ is $\mu_{W1}$-conegligible in
$C([\coint{0,\infty})_0$ (478Ma), so $\Omega'$ is $\mu_W$-conegligible
in $\Omega$.   For $\omega\in\Omega'$,
$h_t(\omega)<\infty$ and $\omega(h_t(\omega))=t$ for every $t\ge 0$.
Of course $h_0(\omega)=0$ for every $\omega\in\Omega$.

If $t\ge 0$ and $\alpha>0$, then

$$\eqalignno{\mu_W\{\omega:\omega\in\Omega,\,
     h_t(\omega)\le\alpha\}&
=\mu_W\{\omega:\omega\in\Omega,\,\max_{s\le\alpha}\omega_0(s)\ge t\}\cr&
=\mu_{W1}\{\omega_0:\omega_0\in C(\coint{0,\infty})_0,\,
     \max_{s\le\alpha}\omega_0(s)\ge t\}
\Displaycause{identifying $\mu_W$ with $\mu_{W1}^2$}
=\Bover{2}{\sqrt{2\pi}}\int_{t/\sqrt{\alpha}}^{\infty}e^{-\xi^2/2}d\xi
\cr}$$

\noindent by 477J.
Accordingly the distribution of $h_t$ is absolutely continuous and
has a Radon-Nikod\'ym derivative, with respect to Lebesgue measure,

\Centerline{$\alpha\mapsto\Bover{d}{d\alpha}\Bover{2}{\sqrt{2\pi}}
   \int_{t/\sqrt{\alpha}}^{\infty}e^{-\eta^2/2}d\eta
=\Bover{2}{\sqrt{2\pi}}\Bover{t}{2\alpha\sqrt{\alpha}}e^{-t^2/2\alpha}
=\Bover{t}{\alpha\sqrt{2\pi\alpha}}e^{-t^2/2\alpha}$}

\noindent for $\alpha>0$.

\medskip

\quad{\bf (iii)} Each $h_t$ is a stopping time adapted to
$\langle\Sigma_s\rangle_{s\ge 0}$ (477I(c-iii)), so we can speak of the
$\sigma$-algebra
$\Sigma_{h_t}=\{E:E\in\Sigma$, $E\cap\{\omega:h_t(\omega)\le s\}\in\Sigma_s$
for every $s\ge 0\}$, as in 612G.

\medskip

{\bf (b)(i)} For $t\ge 0$ and $\omega\in\Omega$, set

$$\eqalign{Z_t(\omega)
&=\omega_1(h_t(\omega))\text{ if }h_t(\omega)<\infty,\cr
&=0\text{ otherwise}.\cr}$$

\noindent Since
$(f,s)\mapsto f(s):C(\coint{0,\infty})\times\coint{0,\infty}\to\Bbb R$ is
continuous for the topology of uniform convergence on compact sets
(4A2G(g-iii)),
$(\omega,s)\mapsto\omega_0(s):\Omega\times\coint{0,\infty}\to\Bbb R$ is
continuous.   Now $h_t$ is lower semi-continuous (477I(c-ii)),
therefore Borel measurable
(4A3Ce), so $Z_t:\Omega\to\Bbb R$ is Borel measurable, therefore
$\Sigma$-measurable.

\medskip

\quad{\bf (ii)} In fact $Z_t$ is always $\Sigma_{h_t}$-measurable.   \Prf\
Take $\alpha\in\Bbb R$ and set $E=\{\omega:Z_t(\omega)>\alpha\}$.   If $s\ge 0$,
$\omega\in E$, $h_t(\omega)\le s$ and $\omega'\in\Omega$ are such that
$\omega'\restr[0,s]=\omega\restr[0,s]$, then
$\omega'_0(h_t(\omega))=\omega_0(h_t(\omega))=t$ and
$\omega'_0(r)=\omega_0(r)<t$ for $r<h_t(\omega)$, so
$h_t(\omega')=h_t(\omega)\le s$,

\Centerline{$Z_t(\omega')=\omega'_1(h_t(\omega))=\omega_1(h_t(\omega))
=Z_t(\omega)>\alpha$,}

\noindent and $\omega'\in E$.   Thus
$E\cap\{\omega:h_t(\omega)\le s\}\in\Sigma_s$;  as $s$ is arbitrary,
$E\in\Sigma_{h_t}$;  as $\alpha$ is arbitrary, $Z_t$ is
$\Sigma_{h_t}$-measurable.\ \Qed

\medskip

\quad{\bf (iii)} For any $t>0$, the distribution of
$Z_t$ is the Cauchy distribution
$\lambda_t$ of
652M.   \prooflet{\Prf\ Take any $a\in\Bbb R$, and set
$f=\chi\ocint{-\infty,a}$.   Then

$$\eqalignno{\mu_W\{\omega:Z_t(\omega)\le a\}&
=\int_{\Omega}fZ_td\mu_W
=\int_{\Omega'}f(\omega_1(h_t(\omega)))\mu_W(d\omega)\cr&
=\int_{\Omega'}f(\omega_1(h_t(\omega_0,\tbf 0)))\mu_W(d\omega)
\Displaycause{where $\tbf 0\in C(\coint{0,\infty})_0$ is the constant
function with value $0$, because $h_t(\omega)$ is calculated from
$\omega_0$}
=\int_{\Omega_0'}\int_{C(\coint{0,\infty})_0}
  f(\omega_1(h_t(\omega_0,\tbf 0)))\mu_{W1}(d\omega_1)\mu_{W1}(d\omega_0)
\Displaycause{identifying $\mu_W$ with $\mu_{W1}^2$}
=\int_{\Omega_0'}\int_{\Bbb R}
    f(\alpha)\Bover1{\sqrt{2\pi h_t(\omega_0,\tbf 0)}}
       e^{-\alpha^2/2h_t(\omega_0,\tbf 0)}
    d\alpha\,\mu_{W1}(d\omega_0)
\Displaycause{by 271Ic,
because $\omega_1\mapsto\omega_1(h_t(\omega_0,\tbf 0))$
is normally distributed with expectation $0$ and variance
$h_t(\omega_0,\tbf 0)$}
=\int_{\Omega_0'}
   \int_{-\infty}^a\Bover1{\sqrt{2\pi h_t(\omega_0,\tbf 0)}}
       e^{-\alpha^2/2h_t(\omega_0,\tbf 0)}
    d\alpha\,\mu_{W1}(d\omega_0)\cr&
=\int_{-\infty}^a\int_{\Omega_0'}
    \Bover1{\sqrt{2\pi h_t(\omega_0,\tbf 0)}}
       e^{-\alpha^2/2h_t(\omega_0,\tbf 0)}
    \mu_{W1}(d\omega_0)d\alpha\cr&
=\int_{-\infty}^a\int_{\Omega'}
    \Bover1{\sqrt{2\pi h_t(\omega_0,\tbf 0)}}
       e^{-\alpha^2/2h_t(\omega_0,\tbf 0)}
    \mu_W(d\omega)d\alpha
\Displaycause{because $\omega\mapsto\omega_0:\Omega\to C(\coint{0,\infty})_0$ is
\imp}
=\int_{-\infty}^a\int_{\Omega'}
    \Bover1{\sqrt{2\pi h_t(\omega)}}
       e^{-\alpha^2/2h_t(\omega)}
    \mu_W(d\omega)d\alpha\cr&
=\int_{-\infty}^a\int_0^{\infty}\Bover1{\sqrt{2\pi\beta}}
       e^{-\alpha^2/2\beta}\Bover{t}{\beta\sqrt{2\pi\beta}}e^{-t^2/2\beta}
    d\beta\,d\alpha
\Displaycause{by (a-ii)}
%=\Bover{t}{2\pi}\int_{-\infty}^a\int_0^{\infty}\Bover1{\beta^2}
%       e^{-(\alpha^2+t^2)/2\beta}d\beta\,d\alpha\cr&
=\Bover{t}{2\pi}
    \int_{-\infty}^a\Bover{2}{\alpha^2+t^2}\int_0^{\infty}
       e^{-\gamma}d\gamma\,d\alpha
\Displaycause{substituting $\gamma=\Bover{\alpha^2+t^2}{2\beta}$}
=\int_{-\infty}^a\Bover{t}{\pi(\alpha^2+t^2)}d\alpha
=\lambda_t\ocint{-\infty,a}.\cr}$$

\noindent As $a$ is arbitrary, we have the result.\ \Qed}

\medskip

\spheader 652Oc Suppose that $0\le t\le t'$.

\medskip

\quad{\bf (i)} By 455U/477G
we have a function $\phi:\Omega\times\Omega\to\Omega$ defined by setting

$$\eqalign{\phi(\omega,\omega')(s)
&=\omega(h_t(\omega))+\omega'(s-h_t(\omega))
  \text{ if } s\ge h_t(\omega),\cr
&=\omega(s)\text{ if }s\le h_t(\omega)\cr}$$

\noindent which is \imp\ for $\mu_W\times\mu_W$ and $\mu_W$.

\medskip

\quad{\bf (ii)} For almost every
$\omega$, $\omega'\in\Omega$, $h_t(\omega)$ and $h_{t'-t}(\omega')$ are
finite, and for such pairs

$$\eqalignno{h_{t'}(\phi(\omega,\omega'))&
=\min\{s:\phi(\omega,\omega')_0(s)\ge t'\}
\Displaycause{writing $\phi(\omega,\omega')_0(s)$ for the first component of
$\phi(\omega,\omega')(s)$}
=\min\{s:s>h_t(\omega),\,\phi(\omega,\omega')_0(s)\ge t'\}
\Displaycause{because if $s\le h_t(\omega)$ then
$\phi(\omega,\omega')_0(s)=\omega_0(s)\le t\}$}
=\min\{s:s>h_t(\omega),\,t+\omega'_0(s-h_t(\omega))\ge t'\}\cr&
=h_t(\omega)+\min\{s:\omega'_0(s)\ge t'-t\}
=h_t(\omega)+h_{t'-t}(\omega'),\cr
Z_{t'}(\phi(\omega,\omega'))&
=\phi(\omega,\omega')_1(h_t(\omega)+h_{t'-t}(\omega'))\cr&
=\omega_1(h_t(\omega))+\omega'_1(h_{t'-t}(\omega'))
=Z_t(\phi(\omega,\omega'))+Z_{t'-t}(\omega').\cr}$$

\noindent Because $\phi$ is \imp,
$Z_{t'}-Z_t$ has the same distribution as

\Centerline{$(\omega,\omega')
  \mapsto Z_{t'}(\phi(\omega,\omega'))-Z_t(\phi(\omega,\omega'))
=Z_{t'-t}(\omega')$}

\noindent which has the same distribution as $Z_{t'-t}$.

\medskip

\quad{\bf (iii)} If $\omega\in E\in\Sigma_{h_t}$ and $\omega'\in\Omega$ is such
that $\omega'\restr[0,h_t(\omega)]=\omega\restr[0,h_t(\omega)]$, then
$\omega'\in E$.   \Prf\ If $h_t(\omega)=\infty$ this is trivial.   Otherwise,
setting $s=h_t(\omega)$, we see that
$\omega\in E\cap\{\tilde\omega:h(\tilde\omega)\le s\}\in\Sigma_s$ while
$\omega'\restr[0,s]=\omega\restr[0,s]$ so
$\omega'\in E\cap\{\tilde\omega:h(\tilde\omega)\le s\}\subseteq E$.\ \Qed

Now if $\omega\in E\in\Sigma_{h_t}$ and $\omega'\in\Omega$ then
$\phi(\omega,\omega')\restr[0,h_t(\omega)]=\omega\restr[0,h_t(\omega)]$ so
$\phi(\omega,\omega')\in E$.   The same applies to the complement of $E$, so
we see that $\phi^{-1}[E]=E\times\Omega$ for every $E\in\Sigma_{h_t}$.
And it follows that $Z_{t'}-Z_t$ is independent of $\Sigma_{h_t}$.   \Prf\
If $E\in\Sigma_{h_t}$ and $\alpha\in\Bbb R$, set
$F=\{\omega:(Z_{t'}-Z_t)(\omega)>\alpha\}$;  then

$$\eqalignno{\mu_W(E\cap F)&
=\mu_W^2(\phi^{-1}[E]\cap\phi^{-1}[F])
=\mu_W^2((E\times\Omega)\cap\phi^{-1}[F])\cr&
=\mu_W^2((E\times\Omega)\cap(\Omega\times\{\omega':Z_{t'-t}(\omega')>\alpha\}))
\Displaycause{by (i) above}
=\mu_W^2((E\times\Omega)\cap(\Omega\times F))
=\mu_WE\cdot\mu_WF.\cr}$$

\noindent As $E$ and $\alpha$ are arbitrary we have the result.\ \Qed

\medskip

\spheader 652Od From (c), we can easily confirm that
$\langle Z_t\rangle_{t\ge 0}$ here has the same distribution as
the process $\langle Z_t\rangle_{t\ge 0}$ of 652N.   But the filtration
$\langle\Sigma_{h_t}\rangle_{t\ge 0}$ is not right-continuous and the paths
$\langle Z_t(\omega)\rangle_{t\ge 0}$ are not \cadlag, so to get a L\'evy
process as described in 652C there is still some work to do.

\medskip

\quad{\bf (i)} If $0\le s\le t$, $E\in\Sigma_{h_s}$ and $r\ge 0$, then

\Centerline{$E\cap\{\omega:h_t(\omega)\le r\}
=(E\cap\{\omega:h_s(\omega)\le r\})\cap\{\omega:h_t(\omega)\le r\}$}

\noindent
is the intersection of two members of $\Sigma_r$ and belongs to $\Sigma_r$;
so $E\in\Sigma_{h_t}$.   Thus $\Sigma_{h_s}\subseteq\Sigma_{h_t}$.

\medskip

\quad{\bf (ii)} For $t\ge 0$, set

\Centerline{$\Tau_t=\{E\symmdiff A:E\in\bigcap_{s>t}\Sigma_{h_s}$,
  $A\in\Cal N(\mu_W)\}$.}

\noindent Then $\langle\Tau_t\rangle_{t\ge 0}$ is a filtration of
$\sigma$-subalgebras of $\Sigma$, and is right-continuous.   \Prf\ If $t\ge 0$ and $E\in\bigcap_{s>t}\Tau_s$, then for each $n\in\Bbb N$ there is
an $A_n\in\Cal N(\mu_W)$ such that
$E\symmdiff A_n\in\bigcap_{s>t+2^{-n}}\Sigma_{h_s}$;  using (i) just above, we
see that
$F=\bigcap_{n\in\Bbb N}E\symmdiff A_n\in\bigcap_{s>t}\Sigma_{h_s}$
and $E\symmdiff F\subseteq\bigcup_{n\in\Bbb N}A_n$ is negligible.\ \QeD\  Note
also that $\Sigma_{h_t}\subseteq\Tau_t$ for every $t$, by (i).

\medskip

\quad{\bf (iii)} For $t\ge 0$ and $\omega\in\Omega$, set
$\tilde h_t(\omega)=\inf_{s>t}h_s(\omega)=\lim_{s\downarrow t}h_s(\omega)$.
Because $h_t\le h_s$ whenever $t\le s$, $h_t\le\tilde h_t$ for every $t$ and
$t\mapsto\tilde h_t(\omega):\coint{0,\infty}\to[0,\infty]$ is
non-decreasing and \cadlag\ for every $\omega$.   We find also that
$\tilde h_t\eae h_t$ for every $t\ge 0$.   \Prf\
$\tilde h_t(\omega)$ is always $\lim_{n\to\infty}h_{t+2^{-n}}(\omega)$, and

\Centerline{$\{\omega:\tilde h_t(\omega)\le\alpha\}
=\bigcap_{m\in\Bbb N}\bigcup_{n\in\Bbb N}
   \{\omega:h_{t+2^{-n}}(\omega)\le\alpha+2^{-m}\}$}

\noindent has measure

\Centerline{$\lim_{m\to\infty}\lim_{n\to\infty}
    \Bover{2}{\sqrt{2\pi}}\int_{(t+2^{-n})/\sqrt{\alpha+2^{-m}}}^{\infty}
       e^{-\xi^2/2}d\xi
=\Bover{2}{\sqrt{2\pi}}\int_{t/\sqrt{\alpha}}^{\infty}e^{-\xi^2/2}d\xi$}

\noindent whenever $t\ge 0$ and $\alpha>0$, by (a-ii) above.   So
$\{\omega:\tilde h_t(\omega)\le\alpha\}$ and $\{\omega:h_t(\omega)\le\alpha\}$
have
the same measure for every $\alpha$;  as the former is always included in the
latter, $\tilde h_t\eae h_t$.\ \Qed

\medskip

\quad{\bf (iv)}
For $t\ge 0$ and $\omega\in\Omega$, set

$$\eqalign{\tilde Z_t(\omega)
&=\omega_1(\tilde h_t(\omega))\text{ if }\omega\in\Omega',\cr
&=0\text{ otherwise}.\cr}$$

\noindent Then $\tilde Z_t\eae Z_t$ for every $t$, and
$\langle\tilde Z_t(\omega)\rangle_{t\ge 0}$ is \cadlag\ for
every $\omega$.   \Prf\ If $\omega\notin\Omega'$ this is trivial.   Otherwise,
$t\mapsto\tilde h_t(\omega):\coint{0,\infty}\to\coint{0,\infty}$ is \cadlag\ and
$\omega_1:\coint{0,\infty}\to\Bbb R$ is continuous so
$t\mapsto\omega_1(\tilde h_t(\omega)):\coint{0,\infty}\to\Bbb R$ is \cadlag.\
\Qed

Next, $\tilde Z_t$ is $\Tau_t$-measurable for every $t\ge 0$.
\Prf\ $\tilde Z_t\eae\lim_{n\to\infty}Z_{t+2^{-n}}$ and
if $s>t$ then $\lim_{n\to\infty}Z_{t+2^{-n}}$ is
$\Sigma_{h_s}$-measurable by (b-ii) above.   So in fact
$\lim_{n\to\infty}Z_{t+2^{-n}}$ is $(\bigcap_{s>t}\Sigma_{h_s})$-measurable
and $\tilde Z_t$ is $\Tau_t$-measurable.\ \Qed

\medskip

\quad{\bf (v)} If $0\le t\le t'$ then $\tilde Z_{t'}-\tilde Z_t$ is independent of
$\Tau_t$.   \Prf\ For $n\in\Bbb N$, $Z_{t'+2^{-n}}-Z_{t+2^{-n}}$ is
independent of $\Sigma_{h_{t+2^{-n}}}$ and therefore of
$\bigcap_{s>t}\Sigma_{h_s}$ and of $\Tau_t$.   Now
$\tilde Z_{t'}-\tilde Z_t\eae\lim_{n\to\infty}Z_{t'+2^{-n}}-Z_{t+2^{-n}}$
so is in the closure of $\{Z_{t'+2^{-n}}-Z_{t+2^{-n}}:n\in\Bbb N\}$ for the
topology of convergence in measure;  by 652Bc, applied with
$\frak B=\{E^{\ssbullet}:E\in\Tau_t\}$, $\tilde Z_{t'}-\tilde Z_t$ is
independent of $\Tau_t$.\ \Qed

\medskip

\quad{\bf (vi)} Putting these together, we see that if $(\frak A,\bar\mu_W)$ is
the measure algebra of $(\Omega,\Sigma,\mu_W)$ and
$\frak B_t=\{E^{\ssbullet}:E\in\Tau_t\}\subseteq\frak A$ for $t\ge 0$,
we have a right-continuous real-time stochastic integration structure
$(\frak A,\bar\mu_W,\coint{0,\infty},\langle\frak B_t\rangle_{t\ge 0})$;
and 613D, applied to $\langle\Tau_t\rangle_{t\ge 0}$
and $\langle\tilde Z_t\rangle_{t\ge 0}$, tells us that
$\langle\tilde Z_t\rangle_{t\ge 0}$ is progressively measurable and corresponds
to a locally near-simple process
$\langle\tilde z_{\sigma}\rangle_{\sigma\in\Cal T_f}$ which is a L\'evy process
in the sense of 652C.
}%end of dvro

%\leader{652P}{}
%\leader{652Q}{}
%\leader{652R}{}
%\leader{652S}{}
%\leader{652T}{}
%\leader{652U}{}
%\leader{652V}{}
%\leader{652W}{}

\exercises{\leader{652X}{Basic exercises (a)}
%\spheader 652Xa
Suppose that \Asissixt\ is a right-continuous real-time
stochastic integration structure and
$\pmb v=\family{\sigma}{\Cal T_f}{v_{\sigma}}$ a L\'evy process.
Show that if $\tau\in\Cal T_f$ and
$\sequencen{t_n}$ is a non-decreasing sequence in $\coint{0,\infty}$, then
$\sequencen{v_{\tau+\check t_{n+1}}-v_{\tau+\check t_n}}$ is independent.
%652H

\spheader 652Xb Show that the formula in 652Ga can be adapted to describe
the sum of any two stopping times, and explore the properties of this
operation.   \Hint{364C.}
%652H

\spheader 652Xc Let $\pmb v=\family{\sigma}{\Cal T_f}{v_{\sigma}}$
be a L\'evy process such that $\alpha=\Expn(v_{\check 1})$ is finite.
Show that $\pmb v-\alpha\pmb\iota$ is a local martingale.   \Hint{632Ma.}
%652K

\spheader 652Xd In 652Ni, show that $\tilde\xi_{ns}$ is a
stopping time adapted to $\langle\Sigma_t\rangle_{t\ge 0}$ for all
$n\in\Bbb N$ and $s>0$.
%652N

\spheader 652Xe In 652Od, show that $\tilde h_t$ is always the
Brownian hitting time
to $\ooint{t,\infty}\times\Bbb R$, so is a stopping time adapted to the
right-continuous filtration $\langle\bigcap_{r>s}\Sigma_r\rangle_{s\ge 0}$.
%652O

\spheader 652Xf Suppose that \Asissixt\ is a real-time stochastic integration
structure.  (i) Show that if $s\ge 0$ then
$\tau\mapsto\tau+\check s:\Cal T\to\Cal T$ is a
lattice homomorphism.   (ii) Show that if $\tau\in\Cal T$ then
$t\mapsto\tau+\check t:\coint{0,\infty}\to\Cal T_f$ is order-continuous.
%652G out of order query

\leader{652Y}{Further exercises (a)}
%\spheader 652Ya
Let $\ddot\mu$ be the measure on $\Cdlg$ corresponding to the
Poisson process, as in 612U.   Show that it is not a Radon measure for the
topology of pointwise convergence on $\Cdlg$.
\leaveitout{ Let $\Omega$ be the support of $\ddot\mu$ (612Ub).   Set
$h(\omega)=\sup\{t:\omega(t)=0\}$ for $\omega\in\Omega$.   Then
$h:\Omega\to[0,\infty]$ is Borel measurable and finite a.e.,
and $h^{-1}[\{t\}]$ is negligible for every $t\ge 0$.
But if $K\subseteq\Omega$ is compact
and $h[K]$ is uncountable, there is a strictly decreasing sequence
$\sequencen{t_n}$ in $h[K]$;  if $\omega_n\in K$ is such that
$h(\omega_n)=t_n$ for each $n$, $\sequencen{\omega_n}$ has no cluster point in
$\Cdlg$.   So $h[K]$ is countable and $K$ is negligible
for every compact $K\subseteq\Omega$.   (In fact $\ddot\mu$ isn't a perfect
measure.)}%\query)

\spheader 652Yb Let $(\frak C,\bar\nu,\coint{0,\infty},
\langle\frak C_t\rangle_{t\ge 0},\Cal T,\family{\tau}{\Cal T}{\frak C_{\tau}})$
be the real-time stochastic integration structure of 612T, and
$\pmb w=\family{\tau}{\Cal T_f}{w_{\tau}}$ Brownian motion.   Set
$\frak A_t=\frak C_{2t}$ for $t\ge 0$, so that
$(\frak C,\bar\nu,\coint{0,\infty},\langle\frak A_t\rangle_{t\ge 0})$ is a
stochastic integration structure, and $\Cal S=\{\check t:t\ge 0\}$.
Show that with respect to the
structure $(\frak C,\bar\nu,\coint{0,\infty},\langle\frak A_t\rangle_{t\ge 0})$,
$\sfamilysS{w}$ is a fully adapted process but is not a local integrator.

\spheader 652Yc Let $f:\coint{0,\infty}\to\Bbb R$ be a \cadlag\ function.
For $t>0$, write $f(t^-)$ for $\lim_{s\uparrow t}f(s)$.   For finite sets
$I\subseteq\coint{0,\infty}$, define $\alpha_I$ by saying that
$\alpha_{\emptyset}=0$ and

\Centerline{$\alpha_I
=\sum_{i=0}^{n-1}f(s_{i+1})-f(s_i)-\med(-1,f(s_{i+1})-f(s_i),1)$}

\noindent if $\#(I)=n>0$ and $\ofamily{i}{n}{s_i}$ is the increasing enumeration
of $I$.   Take $t\ge 0$, and write $Q_t$ for $(\Bbb Q\cap[0,t])\cup\{t\}$.
Show that

\quad (i) For every $\epsilon>0$, $\{s:0<s\le t$, $|f(s)-f(s^-)|\ge\epsilon\}$
is finite;

\quad(ii) $\alpha=\sum_{0<s\le t}f(s)-f(s^-)-\med(-1,f(s)-f(s^-),1)$ is defined;

\quad(iii) For every $\epsilon>0$, there is a finite set $J\subseteq Q_t$
such that
$|\alpha_I-\alpha|\le\epsilon$ whenever $I\subseteq[0,t]$ is finite and
$J\subseteq I$.

\spheader 652Yd Let $(\Omega,\Sigma,\mu)$ be a probability space and
$\langle X_t\rangle_{t\ge 0}$ a family of random variables defined everywhere
on $\Omega$
such that $t\mapsto X_t(\omega):\coint{0,\infty}\to\Bbb R$ is \cadlag\ for every
$\omega\in\Omega$.   For $\omega\in\Omega$ and $t>0$ set
$X_{t^-}(\omega)=\lim_{s\uparrow t}X_s(\omega)$.
For $\omega\in\Omega$ and $t\ge 0$ and non-empty sets
$I\subseteq\coint{0,\infty}$, set

\Centerline{$Y_t(\omega)=\sum_{0<s\le t}X_s(\omega)-X_{s^-}(\omega)
   -\med(-1,X_s(\omega)-X_{s^-}(\omega),1)$,}

\Centerline{$W_I=\sum_{i=0}^{n-1}X_{s_{i+1}}-X_{s_i}
   -\med(-\chi\Omega,X_{s_{i+1}}-X_{s_i},\chi\Omega)$}

\noindent where $\langle s_i\rangle_{i\le n}$ is
the increasing enumeration of $I$.
for $\omega\in\Omega$, write $V(\omega)$ for the total variation of
the function $s\mapsto Y_s(\omega):[0,t]\to\Bbb R$.
Take any $t\ge 0$.   Show that

\quad(i) $Y_t:\Omega\to\Bbb R$ is measurable;

\quad(ii) writing $\Cal I_t$ for the set of
finite subsets of $[0,t]$,
$Y_t^{\ssbullet}=\lim_{I\uparrow\Cal I_t}W_I^{\ssbullet}$
for the topology of convergence in measure on $L^0(\mu)$;

\quad(iii) $V$ is measurable and finite-valued.

\spheader 652Ye Let $\pmb u=\family{\sigma}{\Cal T_f}{u_{\sigma}}$ be a
a classical L\'evy process.   For finite sets $I\subseteq\coint{0,\infty}$
define $w_I$ by saying that $w_I=0$ if $I$ is empty and otherwise
$w_I=\sum_{i=0}^{n-1}u_{\check s_{i+1}}-u_{\check s_i}
   -\med(-\chi 1,u_{\check s_{i+1}}-u_{\check s_i},\chi 1)$ where
$\langle s_i\rangle_{i\le n}$ is the increasing enumeration of $I$.
For $t\ge 0$ let $\Cal I_t$ be the set of finite subsets of $[0,t]$.   Show that

\quad(i) $y_{\check t}=\lim_{I\uparrow\Cal I_t}w_I$ is defined
for every $t\ge 0$;

\quad(ii) $\pmb y=\family{\sigma}{\coint{0,\infty}\varCspcheck}{y_{\sigma}}$
is locally of bounded variation;

\quad(iii) $u_{\check t}-y_{\check t}$ has finite expectation for every
$t\ge 0$;

\quad(iv) $\Expn(u_{\check t}-y_{\check t})=t\Expn(u_{\check 1}-y_{\check 1})$
for every $t\ge 0$.

}%end of exercises

\leader{652Z}{Problem}
%\spheader 652Za
Is the Cauchy process, as described in 652Mc, a local martingale?

\discrversionA{\spheader 652Zb In 652Ni, is $\bigcap_{t\ge 0}\Omega_{1t}$ conegligible?
}{}

\endnotes{
\Notesheader{652} The `classical' L\'evy processes of 652F are the natural
translation of the (real-valued) L\'evy processes of \S455 into the language of
this volume.   What I call a L\'evy process in
652C offers a generalization.   We then find ourselves
doing some work in 652G-652H to show that the processes of 652C
have a weak form of the Markov property of 455U.  For classical L\'evy
processes, it is enough to express the ideas of 633M-633N in terms of \cadlag\
functions (652Yc-652Ye), %652Yc 652Yd 652Ye
giving us an easier route to 652I-652K.   %652I 652J 652K
The calculations
are not trivial, but they are natural once you have come to terms
with the fact that we really do not have continuous sample paths except in very
few cases, and consequently have to understand the jumps
$X_s(\omega)-X_{s^-}(\omega)$, as in 652Yd;  compare 641N.

However, the step from `classical L\'evy process' to `L\'evy process', as
defined here, is more than a formality.   The point is that the
independent-increment property changes from

\inset{whenever $0\le t_0\le\ldots\le t_n$ then $X_{t_1}-X_{t_0},\ldots,
X_{t_n}-X_{t_{n-1}}$ are independent (455Q)}

\noindent to

\inset{$v_{\check t}-v_{\check s}$ is independent of $\frak A_s$ whenever
$0\le s\le t$.}

\noindent The former version translates into

\inset{$v_{\check t}-v_{\check s}$ is independent of
$\{v_{\check s'}:0\le s'\le s\}$ whenever $0\le s\le t$;}

\noindent it comes to the same thing for classical L\'evy processes
because in that context, in effect, $\frak A_s$ is defined as the closed
subalgebra generated by $\{v_{\check s'}:0\le s'\le s\}$.   If we want to move
to more complex structures (for instance, in order to look at an interaction
between two different L\'evy processes, as in multidimensional Brownian motion)
we need to make the distinction.   We cannot take it for granted that a
general L\'evy process $\family{\tau}{\Cal T_b}{v_{\tau}}$ will copy every
feature of the associated classical
L\'evy process defined from the distributions $\lambda_t$ of $v_{\check t}$.
The culminating result of \S455 was the strong Markov property of classical
L\'evy processes (Theorem 455U).   Applied to Brownian motion it was the basis
of large parts of \S\S477-479. %\S477 \S478 \S479
I have not found
a corresponding property of L\'evy processes as defined in 652C.

L\'evy processes have been studied intensively, and in this section I have been
content to stop at the \sm\ property (652K), since this belongs to the basic
classification scheme for stochastic processes which I am following in this
volume.   But you should be aware that the analysis in
652G-652K %652G 652H 652I 652J 652K
amounts to
a few baby steps towards a general description of L\'evy processes, the
L\'evy-Khintchine and L\'evy-It\^o decompositions
({\smc Sato 13}, \S8 and \S\S19-20).
%werner.2016.pdf
%[Sat13] Ken-iti Sato. Lévy processes and infinitely divisible distributions.
%  Cambridge studies in advanced mathematics 68. Cambridge University Press,
%  2013.
%ordered at library 15.7.22
As a very special case, I offer a detailed analysis of the Cauchy distribution
(652M-652O).   % 652M 652N 652O
Regarded as a classical L\'evy process, as in 652M, this is
straightforward enough to have been an exercise in \S455.   %455Xi
But the
L\'evy-It\^o decomposition seeks to describe it in terms of its `L\'evy measure'
$\mu$ in 652N, and to reach the final formula

\Centerline{$\omega(t)=\sum_{n=0}^{\infty}\xi_{nt}(\omega)$ for
$\ddot\mu$-almost every $\omega$}

\noindent in 652Nm I think we must do a good deal more work.
Note that the sum here is a
conditional sum.   We really do have to take the terms in an appropriate order,
biggest jumps first.   Nearly always in this volume we take things as they
arrive, in temporal order.   But in the present case, if $0<s\le t$,
$\omega(s)=\sum_{n=0}^{\infty}\xi_{ns}(\omega)$ is the sum of a subsequence
of $\sequencen{\xi_{nt}(\omega)}$, summing those jumps which occurred at or
before the time $s$, not (for instance) an initial segment.
When we look at the whole jump set $\Jump(\omega)$ for a measure-generic
$\omega\in\Cdlg$, all its points are isolated, but its closure
in $[-\infty,\infty]\times[0,\infty]$ is
$\Jump(\omega)\cup\{(0,t):t\in[0,\infty]\}
\cup\{(\alpha,\infty):\alpha\in[-\infty,\infty]\}$, and it has no useful natural
enumeration.

I dare say you have learnt how to calculate the integrals of 652Mb by contour
integration.   I offer the alternative route through Fourier transforms just
because I gave what I hope was a correct and reasonably complete argument for
the Fourier transform inversion theorem in \S283, and I have omitted contour
integration entirely from this treatise because of the difficulty of presenting
the Jordan Curve Theorem in the style I have chosen.   But if you look at the
contours required for the results here you will have no difficulty in
describing approximating polygons and
triangulations in elementary geometric terms, so that a more or less
elementary version of the Residue Theorem is adequate.

I go through the details of 652O partly because they are striking (and include a
solution to 478Ym\Latereditions), but mostly to offer a different view of the
jumps in the Cauchy process.   I do not think it is obvious from the formula in
652Ma that the Cauchy process is discontinuous (652Md).   In fact the analysis
in 652N shows that it is a `pure jump' process, that is, almost every sample
path is expressible as the sum of its jumps -- provided, that is, that we can
express the sum in the right way, because it is typically not of bounded
variation, and we do not have a saltus function of the type in 226B.   The
method of 652N clearly has a potential for generalization to other probability
measures on $\Bbb R\times\coint{0,\infty}$.   But in the absence of the
general theory of L\'evy processes and L\'evy measures, as described in
{\smc Sato 13},
it is bound to look a bit arbitrary, even though an elementary scaling argument
shows that {\it if} there is an expression in terms of a Poisson point process
as in 652Nc, then the underlying measure must look like the measure $\mu$ of
652Na.   Also, of course, this exposition leans rather heavily on \S495.   On
the other hand, once we have worked through the formulae of 652O, we can see
that the jumps in the paths $\langle\tilde Z_t(\omega)\rangle_{t\ge 0}$ there
correspond to jumps in $g_t(\omega_0)=\inf\{s:\omega_0(s)>t\}$ for
one-dimensional Brownian paths $\omega_0$.   Now we know that almost every
$\omega_0$ is continuous and nowhere differentiable (477K).   It follows that it
will have many local maxima, and
we can expect it to happen (countably and densely)
often that the hitting time to a value $t$ is a strict local maximum of
$\omega_0$, so that $\lim_{s\uparrow t}g_s(\omega_0)$
will be strictly less than $g_t(\omega_0)$.
}%end of notes

\discrpage

\frfilename{mt653.tex}
\versiondate{22.9.20/30.9.23}
\copyrightdate{2014}

\def\chaptername{Applications}
\def\sectionname{Brownian processes}

%\def\varsin{\mathop{\text{s{\i}n}}\nolimits}
\def\highoverline#1{\overline{\vrule height6pt width0pt depth 0pt#1}}

\newsection{653}

In 624F, we saw that the quadratic variation of Brownian motion is the
identity process.   In fact, under suitable conditions, this characterizes
Brownian motion among local martingales (653F).   Elaborating on the
argument, we can show that (again under suitable conditions) general
locally jump-free local martingales can be described in terms of Brownian
motion and a time-change of the type considered in \S635.

\cmmnt{
\leader{653A}{Notation} As always, \Asissix\ will be a stochastic
integration structure;  $L^0=L^0(\frak A)$ will be given its topology of
convergence in measure, defined by the F-norm $\theta$ where
$\theta(w)=\Expn(|w|\wedge\chi 1_{\frak A})$ for $w\in L^0$.
$L^1_{\bar\mu}\subseteq L^0$ will be
the $L$-space defined by $\bar\mu$.

For $t\in T$,
$\check t\in\Cal T$ will be the constant stopping time at $t$;
for $\tau\in\Cal T$, $P_{\tau}$ will be
the conditional expectation associated
with $\frak A_{\tau}$.   If $\Cal S$ is a sublattice of
$\Cal T$, $\Cal I(\Cal S)$ will be the set of finite sublattices of
$\Cal S$.

I will use the symbols $\int$ for Riemann-sum integrals and $\Sint$ for
S-integrals.   If $\Cal S$ is a sublattice of $\Cal T$ then
$\tbf{1}^{(\Cal S)}$ will be the constant process with domain $\Cal S$ and
value $\chi 1_{\frak A}$.   When $T=\coint{0,\infty}$,
$\pmb\iota=\family{\tau}{\Cal T_f}{\iota_{\tau}}$ will be the identity
process.
}

\leader{653B}{Distributions (a)} If $k\ge 1$ and $u_1,\ldots,u_k\in L^0$,
we have a sequentially
order-continuous function $E\mapsto\Bvalue{(u_1,\ldots,u_k)\in E}$
from the Borel
$\sigma$-algebra $\Cal B_k$ of $\Bbb R^k$ to $\frak A$\cmmnt{
(619E)}.   This leads us to a Borel probability measure
$E\mapsto\bar\mu\Bvalue{u\in E}:\Cal B_k\to[0,1]$;  the completion of
this measure is a Radon probability measure $\nu_U$ on $\BbbR^k$\cmmnt{
(256C)}, \cmmnt{which I will call} the {\bf distribution} of
$U=(u_1,\ldots,u_k)$\cmmnt{ (cf.\ 364Yo)}.

\cmmnt{Note that if $(\frak A,\bar\mu)$ is the measure
algebra of a probability space $(\Omega,\Sigma,\mu)$, and the
identification of $L^0(\frak A)$ with $L^0(\mu)$ represents $u_i$ as
$f_i^{\ssbullet}$, where $f_i:\Omega\to\Bbb R$ is measurable for
$1\le i\le k$, then the
distribution $\nu_U$ is just the joint
distribution of the sequence $(f_1,\ldots,f_k)$ of random variables
as described in 271B.}   If $h:\BbbR^k\to\Bbb R$ is a bounded Borel
measurable function, then $\Expn(\bar h(U))=\int h\,d\nu_U$.
\prooflet{\Prf\ Set

$$\eqalign{W&=\{h:h\text{ is a bounded Borel measurable function},\cr
&\mskip100mu
\Expn(\bar h(U))\text{ and }\int h\,d\nu_U\text{ are defined,
finite and equal}\}.\cr}$$

\noindent If $h=\chi E$ where $E\in\Cal B_k$, then $h\in W$
by the definitions of $\bar h$ (619Eb) and $\nu_U$.   Since $W$
is a linear space closed under limits of uniformly bounded
monotonic sequences (use 271E and 619Ef),
it contains all bounded Borel measurable functions.\ \Qed}

\spheader 653Bb We can now speak of the corresponding characteristic
function $\varphi_{\nu_U}$ where

$$\eqalign{\varphi_{\nu_U}(y)
&=\int e^{iy\dotproduct x}\nu_U(dx)
=\int\cos(y\dotproduct x)\nu_U(dx)+i\int\sin(y\dotproduct x)\nu_U(dx)\cr
&=\Expn(\highoverline{\cos}(\eta_1u_1+\ldots+\eta_ku_k))
   +i\Expn(\overline{\sin}(\eta_1u_1+\ldots+\eta_ku_k))\cr}$$

\noindent for $y=(\eta_1,\ldots,\eta_k)\in\BbbR^k$,
\cmmnt{and I will call this} the characteristic function of
$U=(u_1,\ldots,u_k)$\cmmnt{, following 285Ab}.
If now $V\in(L^0)^k$ has the same
characteristic function as $U$, it must have the same distribution as
$U$\cmmnt{, by 285M}.

\spheader 653Bc If $u_1,\ldots,u_k$ are stochastically
independent\cmmnt{ (367W)},
then the distribution $\nu_U$ of $U=(u_1,\ldots,u_k)$ is the product of the
distributions $\nu_{u_i}$ of $u_i$\cmmnt{ (272G)}.

\leader{653C}{Lemma} Let $\Cal S$ be a non-empty sublattice of $\Cal T$
and $\pmb v=\family{\sigma}{\Cal S}{v_{\sigma}}$
a locally jump-free virtually
local martingale such that its quadratic variation
$\pmb v^*$ is an $L^{\infty}$-process.   Writing $v_{\downarrow}$ for the
starting value of $\pmb v$,

\Centerline{$\pmb z_1=\overline{\sin}(\pmb v-v_{\downarrow}\tbf{1}^{(\Cal S)})
\times\highoverline{\exp}(\bover12\pmb v^*)$,
\quad$\pmb z_2=\highoverline{\cos}(\pmb v-v_{\downarrow}\tbf{1}^{(\Cal S)})
\times\highoverline{\exp}(\bover12\pmb v^*)$}

\noindent are martingales.

\proof{ I follow the line of argument in 651B-651E. %651B 651C 651D 651E

\medskip

{\bf (a)} To begin with (down to the end of (c)) I will suppose that $\Cal S$ is
finitely full.   As in 651C, $\pmb v$ is a local integrator; %623Kd
so $v_{\downarrow}$ is defined.  %616Ib 615Gb
As in 651B, $\pmb v^*$ has domain $\Cal S$  %617H 617I
and is a locally jump-free local integrator (617L).  %618T
As in 651D, $\pmb v-v_{\downarrow}\tbf{1}^{(\Cal S)}$
is a virtually local martingale %623Kg
with quadratic variation $\pmb v^*$, so it will be enough to deal with the case
$v_{\downarrow}=0$.

\medskip

{\bf (b)(i)} We can apply 619K
with $h(x,y)=\sin x\exp(\bover12y)$ and $\pmb V=(\pmb v,\pmb v^*)$ to see that

$$\eqalignno{\pmb z_1
=ii_{\bar h\pmb V}(\tbf{1}^{(\Cal S)})
&=\iiv(\highoverline{\cos}(\pmb v)
   \times\highoverline{\exp}(\bsover12\pmb v^*))\cr
&\mskip50mu
+\Bover12 ii_{\pmb v^*}
   (\overline{\sin}(\pmb v)
      \times\highoverline{\exp}(\bsover12\pmb v^*))\cr
&\mskip50mu
-\Bover12 ii_{\pmb v^*}
   (\overline{\sin}(\pmb v)
      \times\highoverline{\exp}(\bsover12\pmb v^*))\cr
&\mskip50mu
+\Bover12 ii_{\covar{\pmb v}{\pmb v^*}}
    (\highoverline{\cos}(\pmb v)
      \times\highoverline{\exp}(\bsover12\pmb v^*))\cr
&\mskip50mu
+\Bover18 ii_{(\pmb v^*)^*}
    (\overline{\sin}(\pmb v)
        \times\highoverline{\exp}(\bsover12\pmb v^*))\cr
&=\iiv(\highoverline{\cos}(\pmb v)
  \times\highoverline{\exp}(\bsover12\pmb v^*))
=\iiv(\pmb z_2)\cr}$$

\noindent because $\covar{\pmb v}{\pmb v^*}$ and
$(\pmb v^*)^*$ are both the zero process, by 624C.

\medskip

\quad{\bf (ii)} Similarly, taking $g(x,y)=\cos x\exp(\bover12y)$,
the starting value of $\pmb z_2=\bar g\pmb V$ is $\chi 1$, so

\Centerline{$\pmb z_2=\tbf{1}^{(\Cal S)}+ii_{\bar g\pmb V}(\tbf{1}^{(\Cal S)})
=\tbf{1}^{(\Cal S)}
  -\iiv(\overline{\sin}(\pmb v)
     \times\highoverline{\exp}(\bover12\pmb v^*))
=\tbf{1}^{(\Cal S)}-\iiv(\pmb z_1)$.}

\medskip

{\bf (c)} By 623O, $\pmb z_1$ and $\pmb z_2$
are virtually local martingales;  because $\Cal S$ is finitely full, they
are approximately local martingales (623K(b-iii)).   Express $\pmb v^*$ as
$\sfamilysS{v^*}$.   If $\tau\in\Cal S$, then there is an $M\ge 0$ such that
$v^*_{\tau}\le M\tbf{1}^{(\Cal S)}$, so that
$\sup(|\pmb z_1|\restr\Cal S\wedge\tau)$
and $\sup(|\pmb z_2|\restr\Cal S\wedge\tau)$
are both at most $e^M\tbf{1}^{(\Cal S)}$.   But this means that
$\pmb z_1\restr\Cal S\wedge\tau$ and $\pmb z_2\restr\Cal S\wedge\tau$
are uniformly integrable;  as $\tau$ is arbitrary, $\pmb z_1$ and $\pmb z_2$ are
martingales (623Na).

\medskip

{\bf (d)} Thus the result is true if $\Cal S$ is finitely full.   In general,
write $\hat\Cal S_f$ for the finitely-covered envelope of $\Cal S$, and
$\hatpmb v$, $\hatpmb v^*$, $\hatpmb z_1$ and $\hatpmb z_2$ for the fully
additive extensions of $\pmb v$, $\pmb v^*$, $\pmb z_1$ and $\pmb z_2$ to
$\hat\Cal S_f$.   As in (a-ii) of the proof of 651E,

\Centerline{$\hatpmb z_1
=\overline{\sin}(\hatpmb v-v_{\downarrow}\tbf{1}^{(\hat\Cal S_f)})
     \times\highoverline{\exp}(\bover12\hatpmb v^*)$,
\quad$\hatpmb z_2
=\highoverline{\cos}(\hatpmb v-v_{\downarrow}\tbf{1}^{(\hat\Cal S_f)})
     \times\highoverline{\exp}(\bover12\hatpmb v^*)$,}

\noindent while $\family{\tau}{\hat\Cal S_f}{\hat v^*_{\tau}}=\hatpmb v^*$
is an $L^{\infty}$-process because if
$\tau\in\hat\Cal S_f$ there are a $\sigma\in\Cal S$ and an $M\ge 0$ such that
$\tau\le\sigma$ and $\hat v^*_{\sigma}\le M\tbf 1^{\hat\Cal S_f}$, in which case
$\hat v^*_{\tau}\le M\tbf 1^{\hat\Cal S_f}$.   So (a)-(c) tell us that
$\hatpmb z_1$ and $\hatpmb z_2$ are martingales, and it follows at once that
$\pmb z_1=\hatpmb z_1\restr\Cal S$ and $\pmb z_2=\hatpmb z_2\restr\Cal S$ are
martingales.
}%end of proof of 653C

\leader{653D}{Lemma} Let $\Cal S$ be a sublattice of $\Cal T$ with least
element $\tau$ and greatest element $\tau'$, and
$\pmb v=\sfamilysS{v}$ a locally jump-free virtually local
martingale with quadratic variation
$\pmb v^*=\sfamilysS{v^*}$.   If $v_{\tau}=0$ and
$v^*_{\tau'}=\gamma\chi 1$ for some $\gamma>0$, then
$v_{\tau'}$ has a normal distribution with mean $0$ and
variance $\gamma$ and is independent of $\frak A_{\tau}$.

\proof{{\bf (a)} Take any non-zero $a\in\frak A_{\tau}$.   Then

$$\eqalignno{\Expn(\chi a\times\overline{\sin}(v_{\tau'}))
&=\Expn(\chi a\times P_{\tau}(\overline{\sin}(v_{\tau'})))
\Displaycause{because $a\in\frak A_{\tau}$}
=e^{-\gamma/2}\Expn(\chi a\times
    P_{\tau}(\overline{\sin}(v_{\tau'})
     \times\highoverline{\exp}(\bsover12v^*_{\tau'})))
\cr&=e^{-\gamma/2}\Expn(\chi a\times\overline{\sin}(v_{\tau})
     \times\highoverline{\exp}(\bsover12v^*_{\tau}))
\Displaycause{653C}
=0,\cr}$$

\noindent and similarly

\Centerline{$\Expn(\chi a\times\highoverline{\cos}(v_{\tau'}))
=e^{-\gamma/2}\Expn(\chi a\times\highoverline{\cos}(v_{\tau}))
=e^{-\gamma/2}\bar\mu a$.}

\noindent Applying the same argument to the virtually
local martingale $\alpha\pmb v$ with
quadratic variation $\alpha^2\pmb v^*$, we see that

\Centerline{$\Expn(\chi a\times\overline{\sin}(\alpha v_{\tau'}))=0$,
\quad$\Expn(\chi a\times\highoverline{\cos}(\alpha v_{\tau'}))
=e^{-\gamma\alpha^2/2}\bar\mu a$}

\noindent for any $\alpha\in\Bbb R$.

\medskip

{\bf (b)} Setting $a=1$, this means that the characteristic function

\Centerline{$\alpha\mapsto\Expn(\highoverline{\cos}(\alpha v_{\tau'}))
   +i\Expn(\overline{\sin}(\alpha v_{\tau'}))$}

\noindent of the distribution of $v_{\tau'}$ is just
$\alpha\mapsto e^{-\gamma\alpha^2/2}$, which is the characteristic function
of the normal distribution with mean $0$ and variance $\gamma$ (285E);
by 285Ma, $v_{\tau'}$ must be normally distributed with mean $0$ and
variance $\gamma$.

\medskip

{\bf (c)} Next,
$\bar\mu(a\Bcap\Bvalue{v_{\tau'}>\beta})
=\bar\mu a\cdot\bar\mu\Bvalue{v_{\tau'}>\beta}$ whenever
$a\in\frak A_{\tau}$ and $\beta\in\Bbb R$.
\Prf\ If $a=0$ this is trivial.
Otherwise, take $\frak B$ to be the principal ideal
of $\frak A$ generated by $a$.   Set
$\bar\nu=\Bover1{\bar\mu a}\bar\mu\restrp\frak B$, and consider
$w\in L^0(\frak B)$ defined by saying that
$\Bvalue{w>\alpha}=a\Bcap\Bvalue{v_{\tau'}>\alpha}$ for every
$\alpha\in\Bbb R$.   Now we see that

\Centerline{$\Expn_{\bar\nu}(\overline{\sin}(\alpha w))
=\Bover1{\bar\mu a}
   \Expn_{\bar\mu}(\chi a\times\overline{\sin}(\alpha v_{\tau'}))
=0$,}

\Centerline{$\Expn_{\bar\nu}(\highoverline{\cos}(\alpha w))
=\Bover1{\bar\mu a}
   \Expn_{\bar\mu}(\chi a\times\highoverline{\cos}(\alpha v_{\tau'}))
=e^{-\gamma\alpha^2/2}$}

\noindent for $\alpha\in\Bbb R$.   But this means that $w$ and $v_{\tau'}$
have the same characteristic function and the same distribution, so that

\Centerline{$\bar\mu(a\Bcap\Bvalue{v_{\tau'}>\beta})
=\bar\mu a\cdot\bar\nu\Bvalue{w>\beta}
=\bar\mu a\cdot\bar\mu\Bvalue{v_{\tau'}>\beta}$.  \Qed}

\noindent As $\beta$ is arbitrary, $v_{\tau'}$ is independent of
$\frak A_{\tau}$, as required.
}%end of proof of 653D

\leader{653E}{Lemma} Let $\Cal S$ be a sublattice of $\Cal T$ with least
element $\tau$ and greatest element $\tau'$, and
$\pmb v=\sfamilysS{v}$ a locally jump-free virtually local
martingale with quadratic variation $\pmb v^*=\sfamilysS{v^*}$, starting
from $v_{\tau}=0$.
If $\tau=\tau_0\le\ldots\le\tau_k$ in $\Cal S$ and
$v^*_{\tau_j}=\gamma_j\chi 1$ for $j\le k$, where
$0=\gamma_0\le\gamma_1\le\ldots\le\gamma_k$, then
$(v_{\tau_0},\ldots,v_{\tau_k})$ has a
centered Gaussian distribution with covariance matrix
$\langle\gamma_{\min(j,l)}\rangle_{j,l\le k}$\cmmnt{
(definition:  456A)}.

\proof{ For $j\le k$, write $w_j$ for $v_{\tau_j}$.

\medskip

{\bf (a)} If $0\le j<k$ and $\gamma_j<\gamma_{j+1}$,
$w_{j+1}-w_j$  has a
normal distribution with mean $0$ and variance $\gamma_{j+1}-\gamma_j$ and
is independent of $\frak A_{\tau_j}$.   \Prf\ Write $\Cal S_j$ for
$\Cal S\cap[\tau_j,\tau_{j+1}]$ and set
$u_{\sigma}=v_{\sigma}-w_j$ for $\sigma\in\Cal S_j$.
Then $\pmb u=\family{\sigma}{\Cal S_j}{u_{\sigma}}$ is
a jump-free local
martingale, being the difference of the jump-free local martingale
$\family{\sigma}{\Cal S_j}{v_{\sigma}}$
and the constant process
$\family{\sigma}{\Cal S_j}{w_j}$.   Also, defining stopping-time
intervals $c(\sigma,\sigma')$ as in 611J and the interval function
$\Delta\pmb u$ as in 613Cc, we have

\Centerline{$(\Delta\pmb u)(c(\sigma,\sigma'))
=u_{\sigma'}-u_{\sigma}
=(\Delta\pmb v)(c(\sigma,\sigma'))$}

\noindent when $\sigma\le\sigma'$ in $\Cal S_j$, so
if we write
$\pmb u^*=\family{\sigma}{\Cal S_j}{u^*_{\sigma}}$
for the quadratic variation of $\pmb u$,

$$\eqalign{u^*_{\tau_{j+1}}
&=\int_{\Cal S_j}(d\pmb u)^2
=\int_{\Cal S_j}(d\pmb v)^2\cr
&=\int_{\Cal S\wedge\tau_{j+1}}(d\pmb v)^2
   -\int_{\Cal S\wedge\tau_j}(d\pmb v)^2
=v^*_{\tau_{j+1}}-v^*_{\tau_j}
=(\gamma_{j+1}-\gamma_j)\chi 1.\cr}$$

\noindent By 653D, $w_{j+1}-w_j=u_{\tau_{j+1}}$ is normally
distributed with mean $0$ and variance $\gamma_{j+1}-\gamma_j$ and
independent of $\frak A_{\tau_j}$.\ \Qed

\medskip

{\bf (b)} On the other hand, if $0\le j<k$ and $\gamma_j=\gamma_{j+1}$,
$w_{j+1}=w_j$, by 624D, so trivially
$w_{j+1}-w_j$ is independent of $\frak A_{\tau_j}$, and in
the language of \S456 has a centered Gaussian distribution with covariance
matrix $(0)$.

\medskip

{\bf (c)} Now $\langle w_{j+1}-w_j\rangle_{j<k}$ is
stochastically independent.   \Prf\ Suppose that $\alpha_j\in\Bbb R$ for
$j<k$.   Then, for each $j<k$,
$\inf_{j'<j}\Bvalue{w_{j'+1}-w_{j'}>\alpha_{j'}}
\in\frak A_{\tau_j}$, so

$$\eqalign{
\bar\mu(\inf_{j'\le j}\Bvalue{w_{j'+1}-w_{j'}>\alpha_{j'}})
&=\bar\mu(\Bvalue{w_{j+1}-w_j>\alpha_j}
 \Bcap\inf_{j'\le j}\Bvalue{w_{j'+1}-w_{j'}>\alpha_{j'}})\cr
&=\bar\mu(\Bvalue{w_{j+1}-w_j>\alpha_{j}})\cdot
 \bar\mu(\inf_{j'\le j}\Bvalue{w_{j'+1}-w_{j'}>\alpha_{j'}}).
 \cr}$$

\noindent Inducing on $j$, we see that

\Centerline{$
\bar\mu(\inf_{j'\le j}\Bvalue{w_{j'+1}-w_{j'}>\alpha_{j'}})
=\prod_{j'\le j}
  \bar\mu(\Bvalue{w_{j'+1}-w_{j'}>\alpha_{j'}})$}

\noindent for $j<k$, and in particular

\Centerline{$
\bar\mu(\inf_{j<k}\Bvalue{w_{j+1}-w_j>\alpha_j})
=\prod_{j<k}
  \bar\mu(\Bvalue{w_{j+1}-w_j>\alpha_j})$.}

\noindent As $\alpha_0,\ldots,\alpha_{k-1}$ are arbitrary,
$\langle w_{j+1}-w_j\rangle_{j<k}$ is
stochastically independent.\ \Qed

\medskip

{\bf (d)} It follows that the distribution of
$\langle w_{j+1}-w_j\rangle_{j<k}$
is the product of the distributions of the
$w_{j+1}-w_j$ (653Bc), and is a centered Gaussian
distribution with covariance matrix
$\langle\beta_{jj'}\rangle_{j,j'<k}$, where
$\beta_{jj}=\gamma_{j+1}-\gamma_j$ and $\beta_{jj'}=0$ if $j\ne j'$ (see
456Be).

\medskip

{\bf (e)} Consequently

\Centerline{$\langle w_j\rangle_{j\le k}
=\langle\sum_{j'=0}^{j-1}w_{j'+1}-w_{j'}\rangle_{j\le k}$}

\noindent has a centered Gaussian distribution (456Ba), and its
covariance matrix is

$$\eqalign{\bigl\langle\Expn(w_j\times w_l)\bigr\rangle_{j,l\le k}
&=\bigl\langle\sum_{j'=0}^{j-1}\sum_{l'=0}^{l-1}
  \Expn((w_{j'+1}-w_{j'})\times(w_{l'+1}-w_{l'}))\bigr\rangle_{j,l\le k}\cr
&=\bigl\langle\sum_{j'=0}^{j-1}\sum_{l'=0}^{l-1}
  \beta_{j'l'}\bigr\rangle_{j,l\le k}\cr
&=\bigl\langle\sum_{j'=0}^{\min(j,l)-1}\gamma_{j'+1}-\gamma_{j'}
  \bigr\rangle_{j,l\le k}
=\langle\gamma_{\min(j,l)}\rangle_{j,l\le k}\cr}$$

\noindent as claimed.
}%end of proof of 653E

\leader{653F}{Theorem}\cmmnt{ (`L\'evy's characterization of Brownian
motion')} Let \Asissixt\
be a right-continuous real-time stochastic integration structure and
$\pmb v=\family{\tau}{\Cal T_f}{v_{\tau}}$
a locally jump-free local martingale such that

\inset{($\alpha$) the quadratic variation of $\pmb v$ is the identity
process,

($\beta$) $\frak A$ is the closed subalgebra of itself defined by
$\{v_{\check t}:t\ge 0\}$,

($\gamma$) for each $t\ge 0$, $\frak A_t$ is the closed subalgebra of
$\frak A$ defined by $\{v_{\check s}:s\le t\}$.}

\noindent
Then $(\frak A,\bar\mu,\coint{0,\infty},\langle\frak A_t\rangle_{t\ge 0},
\Cal T,\family{\tau}{\Cal T}{\frak A_{\tau}},\pmb v)$
is isomorphic to Brownian motion\cmmnt{ as defined in 612T}.

\proof{{\bf (a)} Let
$(\frak C,\bar\nu,\langle\frak C_r\rangle_{r\ge 0},\Cal Q,\pmb w)$
be Brownian motion as described in 612T, writing $\Cal Q$ for the set of
stopping times associated with the filtration
$\langle\frak C_r\rangle_{r\ge 0}$.   For $t\ge 0$ and $\alpha\in\Bbb R$, set
$a_{t\alpha}=\Bvalue{v_{\check t}>\alpha}$ and
$c_{t\alpha}=\Bvalue{w_{\check t}>\alpha}$.   Then

\Centerline{$\bar\mu(\inf_{i\le n}a_{t_i\alpha_i})
=\bar\nu(\inf_{i\le n}c_{t_i\alpha_i})$}

\noindent whenever $t_0,\ldots,t_n\ge 0$ and
$\alpha_0,\ldots,\alpha_n\in\Bbb R$.   \Prf\ Of course we can suppose that
$0=t_0\le\ldots\le t_n$.   In this case, Lemma 653E
tells us that
$(v_{\check t_0},\ldots,v_{\check t_n})$
has a centered Gaussian distribution
with covariance matrix $\langle\min(t_i,t_j)\rangle_{i,j\le n}$.
But of course this is also the distribution of
$(w_{\check t_0},\ldots,w_{\check t_n})$ (477Db, 456Bb).   So

\Centerline{$\bar\mu(\inf_{i\le n}a_{t_i\alpha_i})
=\bar\mu(\inf_{i\le n}\Bvalue{v_{\check t_i}>\alpha_i})
=\bar\mu(\inf_{i\le n}\Bvalue{w_{\check t_i}>\alpha_i})
=\bar\nu(\inf_{i\le n}c_{t_i\alpha_i})$.  \Qed}

\medskip

{\bf (b)} It follows that there is a measure algebra isomorphism
$\phi:\frak C\to\frak A$ such that $\phi(c_{t\alpha})=a_{t\alpha}$ whenever
$t\ge 0$ and $\alpha\in\Bbb R$.   \Prf\ Of course (a) tells us that if
$(s,\alpha)$ and $(t,\beta)$ are such that $c_{s\alpha}=c_{t\beta}$, then
$a_{s\alpha}=a_{t\beta}$.   So (a) can be interpreted as saying that
if $C=\{c_{t\alpha}:r\ge 0$, $\alpha\in\Bbb R\}$, we have a
function $\phi_0:C\to\frak A$ defined by saying that
$\phi_0(c_{t\alpha})=a_{t\alpha}$ for all $t$ and $\alpha$, and
$\bar\nu(\inf_{i\le n}d_i)=\bar\mu(\inf_{i\le n}\phi_0d_i)$ for all
$d_0,\ldots,d_n\in C$.   By 324P, $\phi_0$ has an extension to a
measure-preserving Boolean homomorphism $\phi:\frak D\to\frak A$, where
$\frak D$ is the closed subalgebra of $\frak C$ generated by
$\{c_{t\alpha}:t\ge 0$, $\alpha\in\Bbb R\}$;  but this is
$\frak C$ itself, by 612Td.   Moreover, $\phi[\frak C]$ will be a closed
subalgebra of $\frak A$ including
$\phi_0[C]=\{a_{t\alpha}:t\ge 0$, $\alpha\in\Bbb R\}$ (324Kb), so is the
whole of $\frak A$.   Thus $\phi$ is a surjective measure-preserving
Boolean homomorphism;  being measure-preserving, it is surely injective, so
is an isomorphism.\ \Qed

\medskip

{\bf (c)} Let $T_{\phi}:L^0(\frak C)\to L^0(\frak A)$ be the
$f$-algebra isomorphism corresponding to $\phi$ (612Af).
Because $\phi(c_{t\alpha})=a_{t\alpha}$
for all $t$ and $\alpha$, $T_{\phi}w_{\check t}=v_{\check t}$ for every
$t\ge 0$.

\medskip

{\bf (d)} $\phi[\frak C_t]=\frak A_t$ for every $t\ge 0$.
\Prf\ By 612T(d-ii), $\frak C_t$ is the closed subalgebra defined by
$\{w_{\check s}:s\le t\}$, so
$\phi[\frak C_t]$ is the closed subalgebra defined by
$\{T_{\phi}w_{\check s}:s\le t\}=\{v_{\check s}:s\le t\}$
and is equal to $\frak A_t$.\ \Qed

Thus $\phi$ is an isomorphism between
$(\frak C,\bar\nu,\langle\frak C_t\rangle_{t\ge 0})$ and
$(\frak A,\bar\mu,\langle\frak A_t\rangle_{t\ge 0})$ and therefore
corresponds to an isomorphism between
$(\frak C,\bar\nu,\langle\frak C_t\rangle_{t\ge 0},
\Cal Q,\family{\rho}{\Cal Q}{\frak C_{\rho}})$ and
$(\frak A,\bar\mu,\langle\frak A_t\rangle_{t\ge 0},\Cal T,
\family{\tau}{\Cal T}{\frak A_{\tau}})$.

\medskip

{\bf (e)} It follows that if we define $\hat\phi:\Cal Q\to\Cal T$ as in
634B, so that $\Bvalue{\hat\phi(\rho)>t}=\phi(\Bvalue{\rho>t})$ for
$\rho\in\Cal Q$ and $t\ge 0$, and set

\Centerline{$v'_{\tau}=T_{\phi}w_{\hat\phi^{-1}(\tau)}$}

\noindent for $\tau\in\Cal T_f$, then
$\pmb v'=\family{\tau}{\Cal T_f}{v'_{\tau}}$, like $\pmb w$, will be
locally near-simple.   At the same time, (c) tells us that
if $\Cal T_c$ is the lattice of constant stopping times,
$\pmb v'\restr\Cal T_c=\pmb v\restr\Cal T_c$.   Now $\Cal T_c$ is a
separating cofinal
sublattice of the ideal $\Cal T_b$ of bounded stopping times
(633Da), so $\pmb v'\restr\Cal T_b=\pmb v\restr\Cal T_b$ (633F);  as
$\Cal T_f$ is the covered envelope of $\Cal T_b$,
$\pmb v'=\pmb v$ (612Qa).   Thus

\Centerline{$v_{\tau}=T_{\phi}w_{\hat\phi^{-1}(\tau)}$
for $\tau\in\Cal T_f$,
\quad$v_{\hat\phi(\rho)}=T_{\phi}w_{\rho}$
for $\rho\in\Cal Q_f$,}

\noindent and $\hat\phi$ induces an isomorphism between
$(\frak C,\bar\nu,\langle\frak C_t\rangle_{t\ge 0},\Cal Q,\pmb w)$ and
$(\frak A,\bar\mu,\langle\frak A_t\rangle_{t\ge 0},\Cal T,\pmb v)$.
}%end of proof of 653F

\leader{653G}{Theorem} Suppose that $\family{t}{T}{\frak A_t}$ is
right-continuous.   Let $\Cal S$ be an order-convex sublattice of $\Cal T$
with a least member, $\pmb v=\family{\tau}{\Cal S}{v_{\tau}}$ a
locally jump-free local martingale such that $v_{\min\Cal S}=0$, and
$\pmb v^*=\family{\tau}{\Cal S}{v^*_{\tau}}$ the quadratic
variation of $\pmb v$.
Suppose that for every $n\in\Bbb N$ there is a $\tau\in\Cal S$
such that $v^*_{\tau}\ge n\chi 1_{\frak A}$.   Let
$(\frak C,\bar\nu,\langle\frak C_r\rangle_{r\ge 0},\Cal Q,\pmb w)$
be Brownian motion as described in 612T, again writing $\Cal Q$ for
the set of stopping times associated with $\langle\frak C_r\rangle_{r\ge 0}$.
Express $\pmb w$ as $\family{\sigma}{\Cal Q_f}{w_{\sigma}}$.   Then
there are $\phi$, $\hat\pi$ and $\Cal Q'$ such that

\inset{$\phi:\frak C\to\frak A$ is a
measure-preserving Boolean homomorphism,

$\hat\pi:\Cal Q\to\Cal T$ is a right-continuous lattice homomorphism,

$\Cal Q'=\{\rho:\rho\in\Cal Q_f$, $\hat\pi(\rho)\in\Cal S\}$ is an ideal
in $\Cal Q$ including the ideal $\Cal Q_b$ of bounded stopping times,

taking $T_{\phi}:L^0(\frak C)\to L^0(\frak A)$ to be the
$f$-algebra homomorphism associated with $\phi$ and
$\family{\rho}{\Cal Q_f}{\iota_{\rho}}$ to be the identity process
on $\Cal Q_f$, $v_{\hat\pi(\rho)}=T_{\phi}(w_{\rho})$ and
$v^*_{\hat\pi(\rho)}=T_{\phi}(\iota_{\rho})$ for every $\rho\in\Cal Q'$,

if $\pmb u=\family{\tau}{\Cal S}{u_{\tau}}$ and
$\pmb z=\family{\rho}{\Cal Q'}{z_{\rho}}$ are locally moderately oscillatory
processes such that
$T_{\phi}(z_{\rho})=u_{\hat\pi(\rho)}$ for every $\rho\in\Cal Q'$, then
$\int_{\Cal S\wedge\tau}\pmb u\,d\pmb v
=T_{\phi}(\int_{\Cal Q\wedge\rho}\pmb z\,d\pmb w)$ whenever $\tau\in\Cal S$
and $\rho\in\Cal Q'$ are such that $v^*_{\tau}=T_{\phi}(\iota_{\rho})$.}

\proof{{\bf (a)(i)} Because $\pmb v$ is jump-free, $\pmb v^*$ is jump-free
(618T).   It follows that for every $r\ge 0$ there is a $\tau\in\Cal S$
such that $v^*_{\tau}=r\chi 1_{\frak A}$.  \Prf\ If $r=0$ we can take
$\tau=\min\Cal S$.   Otherwise, by hypothesis, there is a
$\tau'\in\Cal S$ such that $v^*_{\tau'}\ge r\chi 1_{\frak A}$.
By 631Rb, applied to
the jump-free process $\pmb v^*\restr\Cal S\wedge\tau'$, there is
a $\tau\in[\min\Cal S,\tau']$ such that $\|v^*_{\tau}\|_{\infty}\le r$
and $\Bvalue{v^*_{\tau}<r}\Bsubseteq\Bvalue{\tau=\tau'}$;  but this
means that $\Bvalue{v^*_{\tau}<r}=0$ and
$v^*_{\tau}=r\chi 1_{\frak A}$.\ \Qed

\medskip

\quad{\bf (ii)} For $r\ge 0$, set

\Centerline{$\pi_r
=\inf\{\tau:\tau\in\Cal S$, $v^*_{\tau}\ge s\chi 1_{\frak A}$ for some $s>r\}$;}

\noindent because $\Cal S$ is order-convex and has a least element,
$\pi_r\in\Cal S$.   By 632H, $v^*_{\pi_r}\ge r\chi 1_{\frak A}$;  by (i) just
above, $v^*_{\pi_r}\le s\chi 1_{\frak A}$ whenever $r<s$, so
$v^*_{\pi_r}=r\chi 1_{\frak A}$ for every $r\ge 0$.   Of course
$\pi_r\le\pi_s$ whenever $0\le r\le s$, while $\pi_r=\inf_{s>r}\pi_s$ for every
$r\ge 0$, that is, $\langle\pi_r\rangle_{r\ge 0}$ is right-continuous in the
sense of 635Cd.

\medskip

{\bf (b)(i)} For $r\ge 0$ and $\alpha\in\Bbb R$, set
$a_{r\alpha}=\Bvalue{v_{\pi_r}>\alpha}$,
$c_{r\alpha}=\Bvalue{w_{\check r}>\alpha}$.   Then

\Centerline{$\bar\mu(\inf_{i\le n}a_{r_i\alpha_i})
=\bar\nu(\inf_{i\le n}c_{r_i\alpha_i})$}

\noindent whenever $r_0,\ldots,r_n\ge 0$ and
$\alpha_0,\ldots,\alpha_n\in\Bbb R$.   \Prf\ Of course we can suppose that
$r_0\le\ldots\le r_n$.   In this case, Lemma 653E
tells us that
$(0,v_{\pi_{r_0}},\ldots,v_{\pi_{r_n}})$ has a centered Gaussian distribution
with covariance matrix $\langle\min(r_i,r_j)\rangle_{-1\le i,j\le n}$, counting
$0$ as $r_{-1}$.
But of course this is also the distribution of
$(0,w_{\check r_0},\ldots,w_{\check r_n})$.   So

\Centerline{$\bar\mu(\inf_{i\le n}a_{r_i\alpha_i})
=\bar\mu(\inf_{i\le n}\Bvalue{v_{\pi_{r_i}}>\alpha_i})
=\bar\mu(\inf_{i\le n}\Bvalue{w_{\check r_i}>\alpha_i})
=\bar\nu(\inf_{i\le n}c_{r_i\alpha_i})$.  \Qed}

\medskip

\quad{\bf (ii)} It follows that there is a measure-preserving
Boolean homomorphism
$\phi:\frak C\to\frak A$ such that $\phi(c_{r\alpha})=a_{r\alpha}$ whenever
$r\ge 0$ and $\alpha\in\Bbb R$.   \Prf\ Of course (i) tells us that if
$(r,\alpha)$ and $(s,\beta)$ are such that $c_{r\alpha}=c_{s\beta}$, then
$a_{r\alpha}=a_{s\beta}$.   So (i) can be interpreted as saying that
if $C=\{c_{r\alpha}:r\ge 0$, $\alpha\in\Bbb R\}$, we have a
function $\phi_0:C\to\frak A$ defined by saying that
$\phi_0(c_{r\alpha})=a_{r\alpha}$ for all $r$ and $\alpha$, and
$\bar\mu(\phi_0(\inf_{i\le n}d_i))=\bar\nu(\inf_{i\le n}\phi_0(d_i))$ for all
$d_0,\ldots,d_n\in C$.   By 324P again, $\phi_0$ has an extension to a
measure-preserving Boolean homomorphism $\phi:\frak D\to\frak A$, where
$\frak D$ is the closed subalgebra of $\frak C$ generated by
$\{c_{r\alpha}:r\ge 0$, $\alpha\in\Bbb R\}$;  but this is
$\frak C$ itself, by 612Td.\ \Qed

Looking at the associated homomorphism
$T_{\phi}:L^0(\frak C)\to L^0(\frak A)$, we see that

\Centerline{$\Bvalue{T_{\phi}(w_{\check r})>\alpha}
=\phi(\Bvalue{w_{\check r}>\alpha})
=\Bvalue{v_{\pi_r}>\alpha}$}

\noindent whenever $r\ge 0$ and $\alpha\in\Bbb R$, so
$T_{\phi}(w_{\check r})=v_{\pi_r}$ for every $r$.

\medskip

{\bf (c)(i)} Returning to (a-ii), 635B-635C tell us that if we set
$\frak B_r=\frak A_{\pi_r}$ for $r\ge 0$, we have a right-continuous
filtration $\family{r}{\coint{0,\infty}}{\frak B_r}$ of closed subalgebras
of $\frak A$, and that if $\Cal R$ is the lattice of stopping times
adapted to $\family{r}{\coint{0,\infty}}{\frak B_r}$, we have a lattice
homomorphism $\pi:\Cal R\to\Cal T$ such that
$\pi(\check r)=\pi_r$ for every $r\ge 0$ and $\pi$ is
right-continuous.   Note that in the formula
`$\pi(\check r)$' here, $\check r$ must be interpreted as a
constant stopping time in $\Cal R$, which in a formal sense is not
the same thing as a constant stopping time in $\Cal Q$, as examined in (b).
Recall that
$\Bvalue{\sigma=\sigma'}\Bsubseteq\Bvalue{\pi(\sigma)=\pi(\sigma')}$ for all
$\sigma$, $\sigma'\in\Cal R$ (635Cc).

Note next that $\pi(\sigma)\in\Cal S$ for every $\sigma\in\Cal R_b$.
\Prf\ The set
$\Cal R'=\{\sigma:\sigma\in\Cal R$, $\pi(\sigma)\in\Cal S\}$
is an order-convex sublattice of $\Cal R$ (635D(a-i), 635E(h-i)).
We know that $\pi_r\in\Cal S$, so $\check r\in\Cal R'$ for every
$r\ge 0$;  since $\min\Cal R=\check 0$ belongs to $\Cal R'$,
$\Cal R_b\subseteq\Cal R'$.\ \QeD\   Moreover,

\Centerline{$v_{\pi(\check r)}=v_{\pi_r}=T_{\phi}(w_{\check r})$}

\noindent for every $r\ge 0$, by (b-ii).    (I see that I am employing
the notation $\check r$ in both senses here.)

\medskip

\quad{\bf (ii)} If $r\ge 0$ then $\frak C_r$ is the closed subalgebra of
$\frak C$ generated by $\{c_{s\alpha}:s\in[0,r]$, $\alpha\in\Bbb R\}$
(612T(d-ii)).   Consequently $\phi[\frak C_r]$ is the closed subalgebra of
$\frak A$ generated by $\{a_{s\alpha}:s\in[0,r]$, $\alpha\in\Bbb R\}$
(314H) and is included in $\frak A_{\pi_r}$.   Being measure-preserving,
$\phi$ is order-continuous and injective.
We therefore have an injective lattice
homomorphism $\hat\phi:\Cal Q\to\Cal R$ associated with $\phi$ (634B again),
which is order-continuous because
$\family{r}{\coint{0,\infty}}{\frak C_r}$ is right-continuous
(632C(a-i), 634Be).   As noted in 634B(c-i) and 634B(b-iii),
$\Bvalue{\hat\phi(\rho)=\hat\phi(\rho')}=\phi(\Bvalue{\rho=\rho'})$ for all
$\rho$, $\rho'\in\Cal Q$, and
$\hat\phi(\rho)\in\Cal R_f$ whenever $\rho\in\Cal Q_f$.
As in 634Bf, $\iota_{\hat\phi(\rho)}=T_{\phi}(\iota_{\rho})$ for
$\rho\in\Cal Q_f$.

\medskip

{\bf (d)(i)} Set $D=\{\sigma:\sigma\in\Cal R_b$,
$v^*_{\pi(\sigma)}=\iota_{\sigma}\}$.   Then $D=\Cal R_b$.
\Prf\ If $\sigma=\check r$, $\pi(\sigma)=\pi_r$ and
$\iota_{\sigma}=r\chi 1_{\frak A}$, so $\sigma\in D$ by (a-ii).
If $\sigma\in\Cal R_b$ and $\sigma'\in D$, then

\Centerline{$\Bvalue{v^*_{\pi(\sigma)}=\iota_{\sigma}}
\Bsupseteq\Bvalue{v^*_{\pi(\sigma)}=v^*_{\pi(\sigma')}}
  \Bcap\Bvalue{\iota_{\sigma}=\iota_{\sigma'}}
\Bsupseteq\Bvalue{\pi(\sigma)=\pi(\sigma')}
  \Bcap\Bvalue{\sigma=\sigma'}
=\Bvalue{\sigma=\sigma'}$}

\noindent by 635Cc.   So if $\sigma\in\Cal R_b$ belongs to the full
envelope
of $D$, $\sigma\in D$;  in particular, as $\Cal R_b$ is a sublattice of
$\Cal R$, $D$ is finitely full.   If $A\subseteq D$ is non-empty and
downwards-directed and $\sigma=\inf A$, then
$\pi(\sigma)=\inf_{\sigma'\in A}\pi(\sigma')$ and

\Centerline{$v^*_{\pi(\sigma)}
=\inf_{\sigma'\in A}v^*_{\pi(\sigma')}
=\inf_{\sigma'\in A}\iota_{\sigma'}
=\iota_{\sigma}$}

\noindent because $\pmb v^*$ is an order-continuous lattice homomorphism
(632H).   By 633G, $D=\Cal R_b$.\ \Qed

It follows that $\pi$
is injective.   \Prf\ If $\sigma$, $\sigma'$ are distinct members of
$\Cal R$, there is an $r\ge 0$ such that
$\sigma\wedge\check r\ne\sigma'\wedge\check r$;  now

\Centerline{$v^*_{\pi(\sigma)\wedge\pi_r}
=v^*_{\pi(\sigma\wedge\check r)}
=\iota_{\sigma\wedge\check r}
\ne\iota_{\sigma'\wedge\check r}
\ne v^*_{\pi(\sigma')\wedge\pi_r}$}

\noindent so $\pi(\sigma)\wedge\pi_r\ne\pi(\sigma')\wedge\pi_r$
and $\pi(\sigma)\ne\pi(\sigma')$.\ \Qed

\medskip

\quad{\bf (ii)} Set $\hat\pi=\pi\hat\phi:\Cal Q\to\Cal T$.
As both $\hat\phi$ and $\pi$ are injective
right-continuous lattice homomorphisms, so is $\hat\pi$.
For $r\ge 0$, $\hat\pi(\check r)=\pi_r$;  in particular,
$\hat\pi(\min\Cal Q)=\min\Cal S$;  consequently
$\Cal Q'=\hat\pi^{-1}[\Cal S]$ is an
order-convex sublattice of $\Cal Q_f$ containing $\min\Cal Q$, that is,
is an ideal in $\Cal Q_f$.   Also $\Cal Q_b\subseteq\Cal Q'$.

Now $v_{\hat\pi(\rho)}=T_{\phi}(w_{\rho})$ for every $\rho\in\Cal Q_b$.
\Prf\ This time, set $D'=\{\rho:\rho\in\Cal Q_b$,
$v_{\hat\pi(\rho)}=T_{\phi}(w_{\rho})\}$.   If $\rho\in\Cal Q_b$ and
$\rho'\in D'$, then

$$\eqalign{\Bvalue{v_{\hat\pi(\rho)}=T_{\phi}(w_{\rho})}
&\Bsupseteq\Bvalue{v_{\hat\pi(\rho)}=v_{\hat\pi(\rho')}}
  \Bcap\Bvalue{T_{\phi}(w_{\rho})=T_{\phi}(w_{\rho'})}
\cr&\Bsupseteq\Bvalue{\pi(\hat\phi(\rho))=\pi(\hat\phi(\rho'))}
  \Bcap\phi(\Bvalue{w_{\rho}=w_{\rho'}})
\cr&\Bsupseteq\Bvalue{\hat\phi(\rho)=\hat\phi(\rho')}
  \Bcap\phi(\Bvalue{w_{\rho}=w_{\rho'}})
=\phi(\Bvalue{\rho=\rho'}).\cr}$$

\noindent So if $\rho\in\Cal Q_b$ belongs to the full envelope of $D'$,

\Centerline{$\Bvalue{v_{\hat\pi(\rho)}=T_{\phi}(w_{\rho})}
\Bsupseteq\sup_{\rho'\in D'}\phi(\Bvalue{\rho=\rho'})
=\phi(\sup_{\rho'\in D'}\Bvalue{\rho=\rho'})
=\phi(1_{\frak C})=1_{\frak A}$}

\noindent and $v_{\hat\pi(\rho))}=T_{\phi}(w_{\rho})$, that is, $\rho\in D'$.
In particular, $D'$ is finitely full.   By (d), it contains all constant
stopping times.   If $A\subseteq D'$ is non-empty
and downwards-directed and has infimum $\rho$, then
$\{\hat\pi(\rho'):\rho'\in A\}$ is downwards-directed and has infimum
$\hat\pi(\rho)$, so

$$\eqalignno{v_{\hat\pi(\rho)}
&=\lim_{\rho'\downarrow A}v_{\hat\pi(\rho')}
\Displaycause{because $\pmb v$ is locally near-simple}
=\lim_{\rho'\downarrow A}T_{\phi}(w_{\rho'})
=T_{\phi}(\lim_{\rho'\downarrow A}w_{\rho'})
\Displaycause{because $T_{\phi}$ is continuous for the topologies of convergence in
measure on $L^0(\frak C)$ and $L^0(\frak A)$ (613Bn)}
=T_{\phi}(w_{\rho})
\cr}$$

\noindent because $\pmb w$ is locally near-simple.   So $\inf A\in D'$.
Consequently $D'=\Cal Q_b\subseteq D'$, by 633G again.\ \Qed

\medskip

{\bf (e)} We are at last ready to look at some integrals.

\medskip

\quad{\bf (i)} Taking $\pmb u$ and $\pmb z$ as in the last clause of the
statement of this theorem, $S_{\hat\pi[I]}(\pmb u,d\pmb v)
=T_{\phi}(S_I(\pmb z,d\pmb w))$ for every $I\in\Cal I(\Cal Q_b)$.
\Prf\ Because
$\hat\pi\restr I:I\to\hat\phi[I]$ is an injective lattice homomorphism,
$\hat\pi[J]$ will be a maximal
totally ordered subset of $\hat\pi[I]$ whenever $J$ is a maximal totally
ordered
subset of $I$.   So if $\langle\rho_i\rangle_{i\le n}$ linearly generates
the $I$-cells, $\langle\hat\pi(\rho_i)\rangle_{i\le n}$ will linearly generate
the $\hat\pi[I]$-cells (see 611L), and

$$\eqalign{S_{\hat\pi[I]}(\pmb u,d\pmb v)
&=\sum_{i=0}^{n-1}u_{\hat\pi(\rho_i)}
  \times(v_{\hat\pi(\rho_{i+1})}-v_{\hat\pi(\rho_i)})
=\sum_{i=0}^{n-1}T_{\phi}(z_{\rho_i})
  \times(T_{\phi}(w_{\rho_{i+1}})-T_{\phi}(w_{\rho_i}))
\cr&=T_{\phi}(\sum_{i=0}^{n-1}z_{\rho_i}
  \times(w_{\rho_{i+1}}-w_{\rho_i}))
=T_{\phi}(S_I(\pmb z,d\pmb w)).  \text{ \Qed}
\cr}$$

\medskip

\quad{\bf (ii)} Set $\Cal S'=\hat\pi[\Cal Q']$.
Then $\int_{\Cal S'\wedge\hat\pi(\rho)}\pmb u\,d\pmb v
=T_{\phi}(\int_{\Cal Q\wedge\rho}\pmb z\,d\pmb w)$
for every $\rho\in\Cal Q_b$.
\Prf\ Because $\hat\pi:\Cal Q'\to\Cal S$ is an injective lattice
homomorphism, $\hat\pi\restr\Cal Q\wedge\rho$ is a lattice isomorphism
between $\Cal Q\wedge\rho=\Cal Q'\wedge\rho=\Cal Q_b\wedge\rho$
and $\Cal S'\wedge\hat\pi(\rho)$, so

$$\eqalignno{\int_{\Cal S'\wedge\hat\pi(\rho)}\pmb u\,d\pmb v
&=\lim_{J\uparrow\Cal I(\Cal S'\wedge\hat\pi(\rho))}
   S_J(\pmb u,d\pmb v)
=\lim_{I\uparrow\Cal I(\Cal Q\wedge\rho)}
   S_{\hat\pi[I]}(\pmb u,d\pmb v)
\cr&=\lim_{I\uparrow\Cal I(\Cal Q\wedge\rho)}
   T_{\phi}(S_I(\pmb z,d\pmb w))
=T_{\phi}(\lim_{I\uparrow\Cal I(\Cal Q\wedge\rho)}
   S_I(\pmb z,d\pmb w))
=T_{\phi}(\int_{\Cal Q\wedge\rho}\pmb z\,d\pmb w).  \text{ \Qed}\cr
}$$

\medskip

\quad{\bf (iii)} Again suppose that $\rho\in\Cal Q_b$.   Then
$\Cal S'\wedge\hat\pi(\rho)\,\,\pmb v$-separates
$\Cal S\wedge\hat\pi(\rho))$.   \Prf\
Suppose that
$\tau\le\tau'\le\hat\pi(\rho)$ and $v_{\tau}\ne v_{\tau'}$.
Then $v^*_{\tau}\ne v^*_{\tau'}$, by 624E applied to
$\pmb v\restr[\tau,\tau']$.
So there is an $r\ge 0$ such that
$\Bvalue{v^*_{\tau}<r}\Bcap\Bvalue{v^*_{\tau'}>r}\ne 0$.
But $v^*_{\pi_r}=r\chi 1_{\frak A}$, so
$\Bvalue{v^*_{\tau}<v^*_{\pi_r}}
\Bcap\Bvalue{v^*_{\tau'}>v^*_{\pi_r}}\ne 0$.   Since
$\Bvalue{\pi_r\le\tau}\Bsubseteq\Bvalue{v^*_{\pi_r}\le v^*_{\tau}}$
(614Kb),
$\Bvalue{\tau<\pi_r}\Bsupseteq\Bvalue{v^*_{\tau}<v^*_{\pi_r}}$;
similarly,
$\Bvalue{\pi_r<\tau'}\Bsupseteq\Bvalue{v^*_{\pi_r}<v^*_{\tau'}}$.
It follows that, setting $\bar\tau=\pi_r\wedge\hat\pi(\rho)$,
$\Bvalue{\tau\le\bar\tau}\Bcap\Bvalue{\bar\tau<\tau'}\ne 0$.   And

\Centerline{$\bar\tau=\pi_r\wedge\hat\pi(\rho)
=\hat\pi(\check r\wedge\rho)\in\Cal S'\wedge\hat\pi(\rho)$.}

\noindent Thus the conditions of 633Sa are satisfied, and
$\Cal S'\wedge\hat\pi(\rho)\,\,\pmb v$-separates
$\Cal S\wedge\hat\pi(\rho))$.\ \Qed

Of course $\Cal S'$ contains $\hat\pi(\rho)$, so is cofinal with
$\Cal S\wedge\hat\pi(\rho)$, and we can use 633Ka to see that

\Centerline{$T_{\phi}(\int_{\Cal Q\wedge\rho}\pmb z\,d\pmb w)
=\int_{\Cal S'\wedge\hat\pi(\rho)}\pmb u\,d\pmb v
=\int_{\Cal S\wedge\hat\pi(\rho)}\pmb u\,d\pmb v$.}

\medskip

\quad{\bf (iv)} Now suppose that we are given $\rho\in\Cal Q_b$ and
$\tau\in\Cal S$ such that $v^*_{\tau}=T_{\phi}(\iota_{\rho})$,
as in the statement of the
theorem.   Then $v^*_{\tau}=v^*_{\hat\pi(\rho)}$,
so $\pmb v$ is constant
on $[\tau\wedge\hat\pi(\rho),\tau\vee\hat\pi(\rho)]$
(624E) and

\Centerline{$\int_{\Cal S\wedge\tau}\pmb u\,d\pmb v
=\int_{\Cal S\wedge\tau\wedge\hat\pi(\rho)}\pmb u\,d\pmb v
=\int_{\Cal S\wedge\hat\pi(\rho)}\pmb u\,d\pmb v
=T_{\phi}(\int_{\Cal Q\wedge\rho}\pmb z\,d\pmb w)$.}

\medskip

\quad{\bf (v)} Finally, suppose we are told only that $\rho\in\Cal Q'$,
$\tau\in\Cal S$ and $v^*_{\tau}=T_{\phi}(\iota_{\rho})$.   Set
$\tau'=\sup_{r\ge 0}\tau\wedge\pi_r$.   Then

\Centerline{$v^*_{\tau'}
=\sup_{r\ge 0}v^*_{\tau}\wedge v^*_{\pi_r}
=\sup_{r\ge 0}v^*_{\tau}\wedge r\chi 1\frak A
=v^*_{\tau}$,}

\noindent so $\pmb v$ is constant on $[\tau',\tau]$.   Now

\Centerline{$v^*_{\tau\wedge\pi_r}
=v^*_{\tau}\wedge r\chi 1_{\frak A}
=T_{\phi}(\iota_{\rho})\wedge r\chi 1_{\frak A}
=T_{\phi}(\iota_{\rho}\wedge r\chi 1_{\frak C})
=T_{\phi}(\iota_{\rho\wedge\check r})$}

\noindent for each $r$, so

$$\eqalignno{\int_{\Cal S\wedge\tau}\pmb u\,d\pmb v
&=\int_{\Cal S\wedge\tau'}\pmb u\,d\pmb v
=\lim_{r\to\infty}\int_{\Cal S\wedge\tau\wedge\pi_r}\pmb u\,d\pmb v
\Displaycause{because $\iiv(\pmb u)$
is locally jump-free (618R)}
=\lim_{r\to\infty}T_{\phi}(\int_{\Cal Q\wedge\rho\wedge\check r}
   \pmb z\,d\pmb w)
=T_{\phi}(\lim_{r\to\infty}\int_{\Cal Q\wedge\rho\wedge\check r}
   \pmb z\,d\pmb w)
=T_{\phi}(\int_{\Cal Q\wedge\rho}\pmb z\,d\pmb w).
\cr}$$

\noindent This completes the proof.
}%end of proof of 653G

\cmmnt{
\leader{653H}{Remarks} This theorem has been a strenuous journey through a
tangle of technicalities.   It is attempting to say that any locally jump-free
local martingale is a kind of time-changed version of Brownian motion, and
that integration with respect to such a martingale can sometimes
be reduced to integration with respect to Brownian motion.   The key is
in part (b) of the proof.
If we pick the stopping times $\pi_r$ correctly,
we can get a process $\langle v_{\pi_r}\rangle_{r\ge 0}$ which has the
same distribution as $\langle w_r\rangle_{r\ge 0}$ and can therefore be
thought of as a version of Brownian motion.
The measure-preserving homomorphism
$\phi$ is supposed to give a structural foundation for this thought.
Being an injective Boolean homomorphism, it represents $\frak C$ as a
closed subalgebra of $\frak A$.   We have to check that $\frak C_r$ can now
be identified with a subalgebra of $\frak A_{\pi_r}$, and we find that
this identifies the bounded stopping times adapted to
$\family{r}{\coint{0,\infty}}{\frak C_r}$ with a $\pmb v$-separating
sublattice of $\dom\pmb v$, so that the image of $\pmb w$ determines
$\pmb v$.   But here we come
to something important.   $\frak A$ can be a much larger algebra than
$\frak C$.   I do not mean just that it can have arbitrarily large Maharam
type, which is what I usually meant by `large' measure algebra in
Volumes 3 and 5.   From the point of view of the stochastic processes it
supports, what matters more is that there can be large subalgebras of
$\frak A$ which are stochastically independent of the image of
$\frak C$.   For instance, $\frak A$ may have been set up to model a
family of more or less independent martingales -- e.g., Brownian motion in
$\BbbR^n$ where $n>1$ -- and there does not have to be any natural way
of simultaneously reducing them to a single copy of one-dimensional
Brownian motion.

It is this potential complication which makes the final clause on
integration so involved;  while $\phi$, $\hat\pi$ and $\Cal Q'$ all have
constructive definitions, there is no straightforward way of getting
from $\pmb u$ to $\pmb z$ or vice versa, and $\tau$ does not have to be
$\hat\pi(\rho)$.   Given the process $\pmb u$, for instance, we should like
to set $z_{\rho}=T_{\phi}^{-1}u_{\hat\pi(\rho)}$, but this won't work
unless (at least) $u_{\hat\pi(\rho)}\in L^0(\phi[\frak C_{\rho}])$
for enough $\rho$.   (See 653Xb.)
Or given the process $\pmb z$, there will be an
indeterminacy in $\pmb u$ on any interval $[\tau,\tau']$ on which
$\pmb v^*$ (and therefore $\pmb v$) is constant.
Moreover, even if all these difficulties have been resolved, there is no
promise that the required calculations, beginning with the $\pi_r$ of
part (a) of the proof, will be manageable;
and in many important cases, we want to compute an integral
$\int_{\Cal S\wedge\tau}\pmb u\,d\pmb v$ where $\tau$ is unconnected with
$\phi[\frak C]$, and this theorem gives no help at all.

Nevertheless, it tells us that it will always be worth looking at
the possibility of describing a given jump-free martingale in terms of
Brownian motion (see 653Yb), and there are useful special cases in which
simplifications are possible, as in the following.
}%end of comment

\leader{653I}{Corollary} Suppose that $\family{t}{T}{\frak A_t}$ is
right-continuous.   Let $\Cal S$ be an order-convex sublattice of $\Cal T$
with a least member, $\pmb v=\family{\tau}{\Cal S}{v_{\tau}}$ a
locally jump-free local martingale such that $v_{\min\Cal S}=0$, and
$\pmb v^*=\family{\tau}{\Cal S}{v^*_{\tau}}$ the quadratic
variation of $\pmb v$.
Suppose that for every $n\in\Bbb N$ there is a $\tau\in\Cal S$
such that $v^*_{\tau}\ge n\chi 1_{\frak A}$.   Let
$(\frak C,\bar\nu,\family{r}{\coint{0,\infty}}{\frak C_r},\Cal Q,\pmb w)$
be Brownian motion.   Then
there are $\phi$ and $\hat\pi$ such that

\inset{$\phi:\frak C\to\frak A$ is a
measure-preserving Boolean homomorphism,

$\hat\pi:\Cal Q\to\Cal T$ is a lattice homomorphism,

if $f:\Bbb R^2\to\Bbb R$ is continuous, then, taking
$T_{\phi}:L^0(\frak C)\to L^0(\frak A)$ to be the
$f$-algebra homomorphism associated with $\phi$,
$\int_{\Cal S\wedge\hat\pi(\rho)}\bar f(\pmb v,\pmb v^*)\,d\pmb v
=T_{\phi}(\int_{\Cal Q\wedge\rho}\bar f(\pmb w,\pmb\iota)\,d\pmb w)$
whenever $\rho\in\Cal Q_f\cap\hat\pi^{-1}[\Cal S]$.}

\proof{ In 653G, set $\pmb u=\bar f(\pmb v,\pmb v^*)$,
$\tau=\hat\pi(\rho)$ and
$\pmb z=\bar f(\pmb w\restr\Cal Q',\pmb\iota\restr\Cal Q')$,
where $\Cal Q'=\{\rho':\rho'\in\Cal Q_f$, $\hat\pi(\rho')\in\Cal S\}$.
Because $f$ is continuous, these are near-simple, and

$$\eqalignno{T_{\phi}z_{\rho'}
&=T_{\phi}(\bar f(w_{\rho'},\iota_{\rho'}))
=\bar f(T_{\phi}w_{\rho'},T_{\phi}\iota_{\rho'})
\Displaycause{619Eg}
=\bar f(v_{\hat\pi(\rho')},v^*_{\hat\pi(\rho')})
=u_{\hat\pi(\rho')}
\cr}$$

\noindent for every $\rho'\in\Cal Q'$,
so the conditions of 653G are satisfied and we can read the
result off.
}%end of proof of 653I

\leader{653J}{Corollary} Suppose that $\family{t}{T}{\frak A_t}$ is
right-continuous.   Let $\Cal S$ be an order-convex sublattice of $\Cal T$
with a least member, $\pmb v=\family{\tau}{\Cal S}{v_{\tau}}$ a
locally jump-free local martingale such that $v_{\min\Cal S}=0$, and
$\pmb v^*=\family{\tau}{\Cal S}{v^*_{\tau}}$ the quadratic
variation of $\pmb v$.
Suppose that for every $n\in\Bbb N$ there is a $\tau\in\Cal S$
such that $v^*_{\tau}\ge n\chi 1_{\frak A}$.   Let
$(\frak C,\bar\nu,\family{r}{\coint{0,\infty}}{\frak C_r},\Cal Q,\pmb w)$
be Brownian motion.   Then
there are $\phi$ and $\hat\pi$ such that

\inset{$\phi:\frak C\to\frak A$ is a
measure-preserving Boolean homomorphism,

$\hat\pi:\Cal Q\to\Cal T$ is a lattice homomorphism,

if $h:\Bbb R^2\to\Bbb R$ is locally bounded and Borel measurable
then, taking $T_{\phi}:L^0(\frak C)\to L^0(\frak A)$ to be the
$f$-algebra homomorphism associated with $\phi$,
$\Sint_{\Cal S\wedge\hat\pi(\rho)}\bar h(\pmb v,\pmb v^*)\,d\pmb v
=T_{\phi}(\Sint_{\Cal Q\wedge\rho}
  \bar h(\pmb w,\pmb\iota)\,d\pmb w)$
whenever $\rho\in\Cal Q_f\cap\hat\pi^{-1}[\Cal S]$.}

\proof{{\bf (a)}
Take $\phi$, $\hat\pi$, $T_{\phi}$, $\tau$ and $\rho$ as in 653I.
Let $H$ be the set of locally bounded Borel measurable real functions on
$\Bbb R^2$.   Because $\pmb v$ and $\pmb w$ are locally jump-free and
start at zero, they
are equal to their previsible versions $\pmb v_<$ and $\pmb w_<$ (641O)
and are surely locally S-integrable
(directly from the definitions in 645F and 645P), so
$\bar h(\pmb v,\pmb v^*)$ and $\bar h(\pmb w,\pmb\iota)$ are
locally S-integrable (645J) and the S-integrals
$\Sint_{\Cal S\wedge\hat\pi(\rho)}\bar h(\pmb v,\pmb v^*)\,d\pmb v$ and
$\Sint_{\Cal Q\wedge\rho}\bar h(\pmb w,\pmb\iota)\,d\pmb w$ are defined.
Write $H_0$ for the set of those $h\in H$ such that
$\Sint_{\Cal S\wedge\hat\pi(\rho)}\bar h(\pmb v,\pmb v^*)\,d\pmb v
=T_{\phi}(\Sint_{\Cal Q\wedge\rho}\bar h(\pmb w,\pmb\iota)\,d\pmb w)$.

\medskip

{\bf (b)} If $h:\BbbR^2\to\Bbb R$ is continuous, then $h\in H_0$.
\Prf\ $\bar h(\pmb v,\pmb v^*)$ and $\bar h(\pmb w,\pmb\iota)$
are locally jump-free (619Gd), so are equal to
$\bar h(\pmb v,\pmb v^*)_<$ and
$\bar h(\pmb w,\pmb\iota)_<$ except perhaps at $\min\Cal S$, $\min\Cal Q$
respectively;  that is, in the language of \S645,

\Centerline{$(\bar h(\pmb v,\pmb v^*)\restr\Cal S\wedge\hat\pi(\rho))_<
=(\bar h(\pmb v,\pmb v^*)\restr\Cal S\wedge\hat\pi(\rho))
  \times\tbf{1}^{(\Cal S\wedge\hat\pi(\rho))}_<$,}

\Centerline{$(\bar h(\pmb w,\pmb\iota)\restr\Cal Q\wedge\rho)_<
=(\bar h(\pmb w,\pmb\iota)\restr\Cal Q\wedge\rho)
  \times\tbf{1}^{(\Cal Q\wedge\rho)}_<$}

\noindent (using 641G(c-ii) to see that the shift to
$\Cal S\wedge\hat\pi(\rho)$
and $\Cal Q\wedge\rho$ makes no difference).   We therefore have

$$\eqalignno{\Sint_{\Cal S\wedge\hat\pi(\rho)}
  \bar h(\pmb v,\pmb v^*)\,d\pmb v
&=\Sint_{\Cal S\wedge\hat\pi(\rho)}\bar h(\pmb v,\pmb v^*)_<\,d\pmb v
=\int_{\Cal S\wedge\hat\pi(\rho)}\bar h(\pmb v,\pmb v^*)\,d\pmb v
=T_{\phi}(\int_{\Cal Q\wedge\rho}\bar h(\pmb w,\pmb\iota)\,d\pmb w)
\Displaycause{by 653I}
=T_{\phi}(\Sint_{\Cal Q\wedge\rho}\bar h(\pmb w,\pmb\iota)_<\,d\pmb w)
=T_{\phi}(\Sint_{\Cal Q\wedge\rho}\bar h(\pmb w,\pmb\iota)\,d\pmb w)\cr}$$

\noindent and $h\in H_0$.\ \Qed

\medskip

{\bf (c)} If $\sequencen{h_n}$ is a sequence in $H_0$ such that
$h(\alpha)=\lim_{n\to\infty}h_n(\alpha)$ for every
$\alpha\in\Bbb R$ and $\sup_{n\in\Bbb N}|h_n|$ is locally bounded,
then $h\in H_0$.   \Prf\

\Centerline{$\sequencen{
 (\bar h_n(\pmb v,\pmb v^*)\restr\Cal S\wedge\hat\pi(\rho))
    \times\tbfoneminus{\Cal S\wedge\hat\pi(\rho)}}$}

\noindent is uniformly previsibly order-bounded,
because there is a continuous function
$g:\Bbb R\to\coint{0,\infty}$ such that
$|h_n(\alpha,\beta)|\le g(|\alpha|+|\beta|)$ for all $n$, $\alpha$ and
$\beta$ (645Cb), and

$$\eqalign{|(\bar h_n(\pmb v,\pmb v^*)\restr\Cal S\wedge\hat\pi(\rho))
    \times\tbfoneminus{\Cal S\wedge\hat\pi(\rho)}|
&\le(\bar g(|\pmb v|+|\pmb v^*|)\restr\Cal S\wedge\hat\pi(\rho))
    \times\tbfoneminus{\Cal S\wedge\hat\pi(\rho)}
\cr&=(\bar g(|\pmb v|+|\pmb v^*|)\restr\Cal S\wedge\hat\pi(\rho))_<
\cr}$$

\noindent for every $n$.   Also
$\sequencen{\bar h_n(\pmb v,\pmb v^*)\restr\Cal S\wedge\hat\pi(\rho)}$
is order*-convergent
to $\bar h(\pmb v,\pmb v^*)\restr\Cal S\wedge\hat\pi(\rho)$ (642Bd).   So

\Centerline{$\Sint_{\Cal S\wedge\hat\pi(\rho)}
   \bar h(\pmb v,\pmb v^*)\,d\pmb v
=\lim_{n\to\infty}\Sint_{\Cal S\wedge\hat\pi(\rho)}
   \bar h_n(\pmb v,\pmb v^*)\,d\pmb v$}

\noindent by 645T.   Similarly,

\Centerline{$\Sint_{\Cal Q\wedge\rho}\bar h(\pmb w,\pmb\iota)\,d\pmb w
=\lim_{n\to\infty}\Sint_{\Cal Q\wedge\rho}
  \bar h_n(\pmb w,\pmb\iota)\,d\pmb w$.}

\noindent Accordingly

\Centerline{$\Sint_{\Cal S\wedge\hat\pi(\rho)}
    \bar h(\pmb v,\pmb v^*)\,d\pmb v
=\lim_{n\to\infty}
  T_{\phi}(\Sint_{\Cal Q\wedge\rho}\bar h_n(\pmb w,\pmb\iota)\,d\pmb w)
=T_{\phi}(\Sint_{\Cal Q\wedge\rho}\bar h(\pmb w,\pmb\iota)\,d\pmb w)$}

\noindent and $h\in H_0$.\ \Qed

\medskip

{\bf (d)} By 645Cc we see that $H_0=H$, as required.
}%end of proof of 653J

\leader{653K}{Brownian \dvrocolon{processes}}\cmmnt{ The lemmas
653D-653E, and Theorem 653F, have a direct
simplicity which has vanished completely in Theorem 653G and its
corollaries.   In the generality claimed for 653G, with an arbitrary
time-set $T$ and a largely arbitrary locally jump-free local martingale
$\pmb v$, we have to expect complications.   But suppose (as I am sure
many readers are already doing) that $T=\coint{0,\infty}$, that
$\Cal S$ contains all bounded stopping times, and that
$\pmb v^*=\pmb\iota$ is the identity process.   In this case 653E already
tells us what the distribution of $\pmb v$ is on the lattice of constant
stopping times.   Working through the proof of 653G in this case,
we see that

\Centerline{$\iota_{\pi_r}=v^*_{\pi_r}=r\chi 1_{\frak A}
=\iota_{\check r}$}

\noindent so that $\pi_r=\check r$ (here to be interpreted as a member of
$\Cal T$) for every $r\ge 0$.   Consequently $\frak B_r=\frak A_r$ for
every $r$ and $\Cal R=\Cal T$;  moreover, the formula in 635B ensures that
$\pi(\sigma)=\sigma$ for every $\sigma\in\Cal R$.   Accordingly $\hat\pi$,
as defined in (d-ii) of the proof of 653G, is just the natural embedding
$\hat\phi$ of $\Cal Q$ into $\Cal T$ corresponding to the embedding
$\phi:\frak C\to\frak A$ as in 634B.   Because $\phi[\frak C]$, in the
interesting cases, is smaller than $\frak A$, the process $\pmb v$, defined
on an ideal of $\Cal T$, can be
far from isomorphic to the process $\pmb w$.   Nevertheless, it has enough
in common to make it seem useful to have a phrase to describe it.   I will
therefore use the following definition.   Note that (unlike the processes
considered in 653G-653J) I do not require these processes to be strongly
unbounded.   You will see that the abstraction here corresponds to the
difference between `L\'evy process' as defined in 652C and the classical version
in 652F.

\medskip

\noindent}{\bf Definition} A {\bf Brownian-type process} is a
locally jump-free virtually local
martingale $\pmb v$, defined on a lattice $\Cal S$ of stopping times based
on a real-time stochastic integration structure,
such that the quadratic variation of $\pmb v$ is $\pmb\iota\restr\Cal S$.

%\leader{653N}{}
%\leader{653O}{}
%\leader{653P}{}
%\leader{653Q}{}
%\leader{653R}{}
%\leader{653S}{}
%\leader{653T}{}
%\leader{653U}{}
%\leader{653V}{}
%\leader{653W}{}

\exercises{\leader{653X}{Basic exercises (a)}
%\spheader 653Xa
In 653F, show that ($\gamma$) is not a consequence of the
other hypotheses.
%could have $\Bvalue{|v_{\check 1}|\le 1\}\in\frak A_t$ for every $t$.
%653F

\spheader 653Xb In 653G, let $\pmb u=\family{\tau}{\Cal S}{u_{\tau}}$ be a
near-simple fully adapted
process.   Show that there is a fully adapted process
$\pmb z=\family{\rho}{\Cal Q}{z_{\rho}}$ such that
$T_{\phi}(z_{\rho})=u_{\hat\pi(\rho)}$ for every $\rho\in\Cal Q$ iff
$u_{\pi_r}\in L^0(\phi[\frak C_r])$ for every $r\ge 0$, and in this case
$\pmb z$ is near-simple.
%653G

\spheader 653Xc In 653B, show that if $\family{i}{I}{u_i}$ is any family in
$L^0$, there is an associated sequentially order-continuous Boolean homomorphism
$\phi$ from the Baire $\sigma$-algebra $\CalBa(\BbbR^I)$ to $\frak A$ such that
$\phi\{x:x(i)\in E\}=\Bvalue{u_i\in E}$ whenever $i\in I$ and $E$ is a Borel
subset of $\Bbb R$, so that we have a corresponding probability measure
$W\mapsto\bar\mu\phi(W):\CalBa(\BbbR^I)\to[0,1]$ (compare 454J).
%653B

\leader{653Y}{Further exercises (a)}
%\spheader 653Ya
Suppose that
$\family{t}{T}{\frak A_t}$ is right-continuous, that $\Cal S$ is a
sublattice of $\Cal T$ and $\pmb v=\sfamilysS{v}$ is a locally jump-free
local martingale.   Show that there are

\inset{a stochastic integration structure
$(\frak D,\bar\lambda,T^*,\family{t}{T^*}{\frak D_t})$ such that
$T$ is an initial segment of $T^*$ and $T^*\setminus T$ is order-isomorphic
to $\coint{0,\infty}$,

an ideal $\Cal S^*$ of the lattice $\Cal T^*$ of stopping times
adapted to $\family{t}{T^*}{\frak D_t}$,

a locally jump-free local martingale
$\pmb z=\family{\tau}{\Cal S^*}{z_{\tau}}$  such that for every
$r\ge 0$ there is a $\tau\in\Cal S^*$ such that
$z^*_{\tau}\ge r\chi 1_{\frak D}$, where $\pmb z^*$ is the quadratic
variation of $\pmb z$,}

\noindent and
a measure-preserving Boolean homomorphism $\phi:\frak A\to\frak D$
such that $z_{\pi(\sigma)}$ is defined and equal to
$T_{\phi}(v_{\sigma})$ for every $\sigma\in\Cal S$, where for
$\sigma\in\Cal T$

$$\eqalign{\Bvalue{\pi(\sigma)>t}
&=\phi(\Bvalue{\sigma>t})\text{ if }t\in T,\cr
&=0\text{ if }t\in T^*\setminus T,\cr}$$

\noindent and $T_{\phi}:L^0(\frak A)\to L^0(\frak D)$ is the $f$-algebra
homomorphism defined from $\phi$.   Show that this can be done in such
a way that $\pi[\Cal S]\,\,\pmb z$-separates $\Cal S^*$.
%653J

\spheader 653Yb Use 653Ya and 653G to show that 651F implies 651C.
%653J
}%end of exercises

\endnotes{
\Notesheader{653} Clearly 653C is just the complex-valued version of
651C expressed in terms of its real and imaginary parts separated.
We could omit this
step altogether, and go straight from 651C to 653D, if we re-worked
the theory so far with the complex linear space $L^0_{\Bbb C}$ (241J, 366M)
in place of the real linear space $L^0$.

The argument for 653F depends heavily on the fact that we can define
an isomorphism of a structure
$(\frak A,\bar\mu,\langle\frak A_t\rangle_{t\ge 0},\Cal T,\pmb v)$
in terms of a measure-preserving Boolean isomorphism.   I have tried to
present this neither too pedantically nor too glibly.
The point of the careful elaboration of Chapter 61
is to show how the processes we are looking at are definable
in terms of stochastic integration structures \Asisfour\
as described in the notes to \S613, so that isomorphisms of these
structures will carry faithful copies of the processes with them.

You will see that the main results of \S635 are an
attempt to generalize part of the proof of 653G.   A conspicuous
difficulty in applying 653G is the requirement that the
quadratic variation of the martingale $\pmb v$ should be unbounded in a
strong sense.   However this is not the real obstacle, since with a
different kind of time-change any martingale has a strongly unbounded
end-extension (653Ya).

In 653I-653J I look at functions of two variables and processes
$\pmb u=\bar f(\pmb v,\pmb v^*)$, $\pmb z=\bar f(\pmb w,\pmb\iota)$.
In fact what really matters is that $\pmb u$ is calculated from
$\pmb v$ by the same `legitimate'
method as $\pmb z$ is calculated from $\pmb w$.
Introducing the quadratic variations $\pmb v^*$ and $\pmb w^*=\pmb\iota$
is permitted because

\Centerline{$\pmb v^*=\pmb v^2-2\iiv(\pmb v)$,
\quad$\pmb\iota=\pmb w^2-2ii_{\pmb w}(\pmb w)$}

\noindent and indefinite integration is a legitimate method.   But to go
farther with this idea we should need a definition of
`legitimate method' which included a way of matching operations in
different stochastic integration structures.
}%end of notes

\discrpage



\frfilename{mt654.tex}
\versiondate{8.12.16/8.9.23}
\copyrightdate{2014}

\def\sm{semi-martingale} %hyphen to remind me that it's the "virtual" kind

\def\chaptername{Applications}
\def\sectionname{Picard's theorem}

\def\1lim{\mathop{\text{1lim}}}

\newsection{654}

The general theory of solutions of ordinary differential equations begins
with a classical existence and uniqueness theorem:  if $h$ is a continuous
function of two variables which is Lipschitz in the first variable, then
the differential equation

\Centerline{$x'(t)=h(x(t),t)$,\quad$x(0)=x_{\star}$}

% x^*  unsatisfactory as we have quadratic variations later

\noindent or, equivalently, the integral equation

\Centerline{$x(t)=x_{\star}+\int_0^th(x(s),s)ds$}

\noindent has a unique solution.   In this section I present corresponding
results for stochastic
integral equations of this type, first for the Riemann-sum
integral (654I) and then for the S-integral (654N).

\cmmnt{
\leader{654A}{Notation} \Asissix\ will be a
stochastic integration structure,
and $\Expn$ the integral on $L^1(\frak A,\bar\mu)$;   $L^0$, $L^2$,
$L^{\infty}$ and $\theta$ will be
$L^0(\frak A)$, $L^2(\frak A,\bar\mu)$, $L^{\infty}(\frak A)$
and the standard F-norm
defining the topology of convergence in measure on $L^0$ (613B).

For a sublattice $\Cal S$ of
$\Cal T$ and $\tau\in\Cal S$, I write
$\Cal S\wedge\tau$ for $\{\sigma:\sigma\in\Cal S$, $\sigma\le\tau\}$,
$\Cal S\vee\tau$ for $\{\sigma:\sigma\in\Cal S$, $\tau\le\sigma\}$ and
$\Cal I(\Cal S)$ for the upwards-directed set of finite sublattices of
$\Cal S$.  $\tbf 1^{(\Cal S)}$ will be the
constant process with domain $\Cal S$ and value $\chi 1$.

If $\pmb u=\sfamilysS{u}$ is a process,
$\sup|\pmb u|=\sup_{\sigma\in\Cal S}|u_{\sigma}|$ if this is defined in
$L^0$, and
$\|\pmb u\|_{\infty}=\sup_{\sigma\in\Cal S}\|u_{\sigma}\|_{\infty}$,
counting this as $\infty$ if any $u_{\sigma}$ does not belong to
$L^{\infty}$.   If $\pmb u=\sfamilysS{u}$ is a process and
$z\in L^0(\frak A\cap\bigcap_{\sigma\in\Cal S}\frak A_{\sigma})$, then
$z\pmb u=\sfamilysS{z\times u}$.   If $\pmb u=\sfamilysS{u}$ and
$\pmb v=\sfamilysS{v}$ are processes, then
$\Bvalue{\pmb u\ne\pmb v}
=\sup_{\sigma\in\Cal S}\Bvalue{u_{\sigma}\ne v_{\sigma}}$.

If $h:\BbbR^k\to\Bbb R$ is Borel measurable, then I write
$\bar h$ for any of the corresponding functions from $(L^0)^k$ to $L^0$ and
from $((L^0)^{\Cal S})^k$ to $(L^0)^{\Cal S}$ for a set
$\Cal S$, as in 619E-619G. %619E 619F 619G

If $\Cal S$ is a sublattice of $\Cal T$, $\Mob(\Cal S)$ is the space of
order-bounded fully adapted processes with domain $\Cal S$,
$\Mns(\Cal S)$ the space of near-simple processes with domain $\Cal S$,
$\Mmo(\Cal S)$ the space of moderately oscillatory processes with domain
$\Cal S$, and for $\pmb u\in\Mmo(\Cal S)\,\,\pmb u_<$ is its previsible
version.   $\Mpob(\Cal S)$ is
the solid linear subspace of $\Mob(\Cal S)$ generated by
$\{\pmb u_<:\pmb u\in\Mns(\Cal S)\}$.   The ucp topology on $\Mob(\Cal S)$
is the linear space topology defined by the F-norm
$\pmb u\mapsto\theta(\sup|\pmb u|)$ as in 615B.
$\MSi^0(\Cal S)$ is the closure of
$\{\pmb u_<:\pmb u\in\Mns(\Cal S)\}$ in $\Mpob(\Cal S)$ for the
S-integration topology (645E-645F), and
$\MSi(\Cal S)=\{\pmb x:\pmb x\in\Mob(\Cal S)$,
$\pmb x\times\tbfoneminus{\Cal S}\in\MSi^0(\Cal S)\}$.

$S_I(\pmb u,d\pmb v)$ will denote a Riemann sum (613Fb);
$\int_{\Cal S}\pmb u\,d\pmb v$ and $\int_{\Cal S}\pmb u\,|d\pmb v|$ will be
Riemann-sum integrals (613H), with associated indefinite integrals
$\iivu$ (613O);  and $\Sint_{\Cal S}\pmb x\,d\pmb v$ and
$\Siiv(\pmb x)$ will
refer to the S-integral and indefinite S-integral defined in 645P and 646K.
}%end of cmmnt

\leader{654C}{Lemma} (a) Let $\Cal S$ be a non-empty
sublattice of $\Cal T$ such that
$\inf_{\tau\in\Cal S}\sup_{\sigma\in\Cal S}\Bvalue{\tau<\sigma}=0$, and
$\pmb u=\sfamilysS{u}$ a fully adapted process.

(a) If $\pmb u$ is locally order-bounded, it is order-bounded.

(b) If $\pmb u$ is locally moderately oscillatory, it is moderately oscillatory.

(c) If $\pmb u$ is locally near-simple, then it is near-simple.

\proof{ For $\tau\in\Cal S$, set
$a_{\tau}=\sup_{\sigma\in\Cal S}\Bvalue{\tau<\sigma}$.   Then
$a_{\tau'}\Bsubseteq a_{\tau}$ whenever $\tau\le\tau'$ in $\Cal S$,
while we are supposing that $\inf_{\tau\in\Cal S}a_{\tau}=0$, so
$\inf_{\tau\in\Cal S}\bar\mu a_{\tau}=0$.

\medskip

{\bf (a)} Suppose that $\pmb u$ is locally order-bounded.
Set $A=\{u_{\sigma}:\sigma\in\Cal S\}$.   Let $\epsilon>0$.   Then there is a
$\tau\in\Cal S$ such that $\bar\mu a_{\tau}\le\epsilon$.   For any
$\sigma\in\Cal S$,

\Centerline{$\Bvalue{u_{\sigma}\ne u_{\sigma\wedge\tau}}
\Bsubseteq\Bvalue{\tau<\sigma}\Bsubseteq a_{\tau}$,}

\noindent so $u_{\sigma}\times\chi(1\Bsetminus a_{\tau})=
u_{\sigma\wedge\tau}\times\chi(1\Bsetminus a_{\tau})$.   Accordingly

\Centerline{$\{u\times\chi(1\Bsetminus a_{\tau}):u\in A\}
\Bsubseteq\{u_{\sigma}\times\chi(1\Bsetminus a_{\tau}):
\sigma\in\Cal S\wedge\tau\}$}

\noindent is order-bounded, because $\pmb u$ is locally order-bounded,
while $\bar\mu(1\Bsetminus a_{\tau})\ge 1-\epsilon$.   By 613B(p-i),
$A$ is order-bounded in $L^0$, that is, $\pmb u$ is an order-bounded process.

\medskip

{\bf (b)} Take $\epsilon>0$.   Let $\tau\in\Cal S$ be such that
$\bar\mu a_{\tau}\le\epsilon$.   Set $\pmb u'=\sfamilysS{u'}$ where
$u'_{\sigma}=u_{\sigma\wedge\tau}$ for $\sigma\in\Cal S$.
Then $\pmb u'$ is fully adapted (612Ib).
$\pmb u'\restr\Cal S\wedge\tau$ is moderately oscillatory because it is equal to
$\pmb u\restr\Cal S\wedge\tau$, and $\pmb u'\restr\Cal S\vee\tau$ is moderately
oscillatory because it is constant.   So $\pmb u'$ is moderately oscillatory
(615F(a-v)).   If $\sigma\in\Cal S$,

\Centerline{$\Bvalue{u'_{\sigma}\ne u_{\sigma}}
=\Bvalue{u_{\sigma\wedge\tau}\ne u_{\sigma}}
\Bsubseteq\Bvalue{\sigma\wedge\tau<\sigma}
=\Bvalue{\tau<\sigma}\Bsubseteq a_{\tau}$,}

\noindent while $\pmb u$ is order-bounded by (a) above, so

\Centerline{$\sup|\pmb u-\pmb u'|\le 2\sup|\pmb u|\times\chi a_{\tau}$}

\noindent and $\theta(\sup|\pmb u-\pmb u'|)\le\bar\mu a_{\tau}\le\epsilon$.
As $\epsilon$ is arbitrary,
$\pmb u$ belongs to the topological closure of $\Mmo(\Cal S)$ in
$\Mob(\Cal S)$ and is moderately oscillatory (615F(a-iv)).

\medskip

{\bf (c)} If $\pmb u$ is locally near-simple, then we it is locally moderately
oascillatory (by 631Ca) and moderately oscillatory (by (b) above), so is
near-simple by 631F(c-ii).
}%end of proof of 654C

\leader{654D}{Lemma} Let $\Cal S$ be a sublattice of $\Cal T$
and $h:\BbbR^k\to\Bbb R$ a continuous function.   Then
$\bar h(\pmb u_1,\ldots,\pmb u_k)\in\Mob=\Mob(\Cal S)$ whenever
$\pmb u_1,\ldots,\pmb u_k\in\Mob$, and
$\bar h:\Mob^k\to\Mob$ is continuous for the ucp topology on $\Mob$.

\leaveitout{
(c) If every $\pmb u_i$ is near-simple and $h$ is continuous and $h(0)=0$,
then $\bar h(\pmb u_1,\ldots,\pmb u_k)_<
=\bar h(\pmb u_{1-},\ldots,\pmb u_{k-})$.
%maybe not needed
}

\proof{ We can repeat the argument of 615Ca.   The result is trivial if
$\Cal S$ is empty;  suppose otherwise.   Express
$\pmb u_i$ as $\family{\sigma}{\Cal S}{u_{i\sigma}}$ for each $i$.

\medskip

{\bf (a)} For $\xi\in\Bbb R$, set
$g(\xi)=\sup\{|h(x)|:x\in\Bbb R^k$, $\|x\|_{\infty}\le|\xi|\}$;  then
$g$ is continuous, and $|\bar h(v_1,\ldots,v_k)|\le\bar g(v)$
whenever $v_1,\ldots,v_k,v\in L^0$
and $|v_i|\le v$ for $1\le i\le k$.

Suppose that $\pmb{u}_i=\family{\sigma}{\Cal S}{u_{i\sigma}}$
belongs to $\Mob$ for $1<i\le k$.
Set $\bar u=\sup_{1\le i\le k}\sup|\pmb u_i|$.
Then $|\bar h(u_{1\sigma},\ldots,u_{k\sigma})|\le g(\bar u)$
for every $\sigma\in\Cal S$, so
$\{\bar h(u_{1\sigma},\ldots,u_{k\sigma}):\sigma\in\Cal S\}$
is order-bounded in $L^0$, and $\bar h(\pmb{u}_1,\ldots,\pmb u_k)\in\Mob$.

\medskip

{\bf (b)} Now take
$\pmb{v}_i=\family{\sigma}{\Cal S}{v_{i\sigma}}\in\Mob$ for $1\le i\le k$
and $\epsilon>0$.
Set $\bar v=\sup_{\sigma\in\Cal S,1\le i\le k}|v_{i\sigma}|$,
and let $M\ge 0$ be such that $\bar\mu\Bvalue{\bar v>M}\le\epsilon$.
Let $\delta\in\ocint{0,1}$ be such that
$|h(x)-h(y)|\le\epsilon$ whenever $y\in[-M-1,M+1]^k$ and
$\|x-y\|_{\infty}\le\delta$.   Then for any $w_1,\ldots,w_k$,
$w'_1,\ldots,w'_k\in L^0$,

$$\eqalign{&\Bvalue{|\bar h(w'_1,\ldots,w'_k)-\bar h(w_1,\ldots,w_k)|
  >\epsilon}
\cr&\mskip100mu
\Bsubseteq\sup_{1\le i\le k}\Bvalue{|w_i|>M}\Bcup\Bvalue{|w'_i-w_i|>\delta}.
\cr}$$

Take any $\pmb{u}_1,\ldots,\pmb u_k\in\Mob$ such that
$\theta(\sup|\pmb{u}_i-\pmb{v}_i|)\le\delta\epsilon$ for every
$i$.   Set $\bar u
=\sup_{\sigma\in\Cal S,1\le i\le k}\penalty-100|u_{i\sigma}-v_{i\sigma}|$
and
$\bar w=\sup_{\sigma\in\Cal S}|\bar h(u_{1\sigma},\ldots,u_{k\sigma})
  -\bar h(v_{1\sigma},\ldots,v_{k\sigma})|$.
Then $\bar\mu\Bvalue{\bar u>\delta}\le k\epsilon$, so

$$\eqalign{\Bvalue{\bar w>\epsilon}
&=\sup_{\sigma\in\Cal S}
   \Bvalue{|\bar h(u_{1\sigma},\ldots,u_{k\sigma})
       -\bar h(v_{1\sigma},\ldots,v_{k\sigma})|>\epsilon}\cr
&\Bsubseteq\sup_{\sigma\in\Cal S,1\le i\le k}\Bvalue{|v_{i\sigma}|>M}
   \Bcup\sup_{\sigma\in\Cal S,i\le k}
      \Bvalue{|u_{i\sigma}-v_{i\sigma}|>\delta}
=\Bvalue{\bar v>M}\Bcup\Bvalue{\bar u>\delta}\cr}$$

\noindent has measure at most $(k+1)\epsilon$, and

\Centerline{$\theta(\sup
  |\bar h(\pmb{u}_1,\ldots,\pmb u_k)-\bar h(\pmb{v}_1,\ldots,\pmb v_k)|)
=\theta(\bar w)
\le (k+2)\epsilon$.}

\noindent As $\pmb{v}_1,\ldots,\pmb v_k$ and $\epsilon$ are arbitrary,
$\bar h:\Mob^k\to\Mob^k$ is continuous.
}%end of proof of 654D

\leader{654E}{Lemma} Let $\Cal S$ be a sublattice of $\Cal T$ with a greatest
element, and define $\pmb z=\sfamilysS{z}$ by setting
$z_{\sigma}=\chi\Bvalue{\sigma<\max\Cal S}$ for $\sigma\in\Cal S$.

(a) Suppose that
$\pmb u\cmmnt{\mskip5mu=\sfamilysS{u}}$ is a moderately oscillatory process and
$\pmb v\cmmnt{\mskip5mu=\sfamilysS{v}}$ an integrator.   Then

\Centerline{$\pmb z\times\iivu
=\pmb z\times ii_{\pmb v}(\pmb z\times\pmb u)
=\pmb z\times ii_{\pmb z\times\pmb v}(\pmb u)$}

\noindent and $\int_{\Cal S}\pmb u\,d\pmb v
=\int_{\Cal S}\pmb z\times\pmb u\,d\pmb v$.

(b) Let $\pmb w\cmmnt{\mskip5mu=\sfamilysS{w}}$
be a process of bounded variation, and
$\pmb w^{\uparrow}\cmmnt{\mskip5mu=\sfamilysS{w^{\uparrow}}}$
its cumulative variation\cmmnt{ (614O)}.
Then $\int_{\Cal S}|d(\pmb z\times\pmb w)|
\le\sup|\pmb z\times(\pmb w^{\uparrow}+|\pmb w|)|$.

(c) Suppose that $\Cal S$ has a least member and
that $\pmb w=\sfamilysS{w}$ is an order-bounded fully adapted process
starting from $w_{\min\Cal S}=0$.
Then $\sup|\pmb w|
\le\sup|\pmb z\times\pmb w|+\Osclln(\pmb w)$\cmmnt{, where
$\Osclln(\pmb w)$ is the residual oscillation of $\pmb w$ (618B)}.

(d) Suppose that
$\pmb w=\sfamilysS{w}$ is a fully adapted process and that
$\alpha\ge 0$ is such that
$\Bvalue{\sigma<\max\Cal S}\Bsubseteq\Bvalue{|w_{\sigma}|\le\alpha}$ for
$\sigma\in\Cal S$.   Then $\pmb w$ is order-bounded,
$\|\sup|\pmb z\times\pmb w|\|_{\infty}\le\alpha$
and $\|\pmb w\|_{\infty}\le\alpha+\|\Osclln(\pmb w)\|_{\infty}$.

(e) If $\pmb u\in\Mmo(\Cal S)$ then $\pmb u_<=(\pmb z\times\pmb u)_<$.

\proof{{\bf (a)(i)} I should begin by pointing out that $\pmb z$ is the simple
process with domain $\Cal S$, breakpoint string $(\max\Cal S)$, starting value
$\chi 1$ and value $0$ at $\max\Cal S$.   So in particular it is of bounded
variation (614Q(a-iii), or otherwise), hence an integrator (616Ra);
consequently
$\pmb z\times\pmb u$ is an integrator (616Pa, or otherwise) and moderately
oscillatory (616Ib).   Thus all the indefinite integrals here have domain
$\Cal S$.

Take $\sigma\in\Cal S$ and set $a=\Bvalue{\sigma<\max\Cal S}$.   If
$I\in\Cal I(\Cal S\wedge\sigma)$ then

\Centerline{$S_I(\pmb u,d\pmb v)\times\chi a
=S_I(\pmb z\times\pmb u,d\pmb v)\times\chi a
=S_I(\pmb u,d(\pmb z\times\pmb v))\times\chi a$.}

\noindent\Prf\ If $I$ is empty, this is trivial.   Otherwise, take
$(\tau_0,\ldots,\tau_n)$ linearly generating the $I$-cells.   Then

$$\eqalignno{S_I(\pmb u,d\pmb v)\times\chi a
&=\sum_{i=0}^{n-1}u_{\tau_i}\times(v_{\tau_{i+1}}-v_{\tau_i})\times\chi a
\cr&=\sum_{i=0}^{n-1}u_{\tau_i}\times z_{\tau_i}
  \times(v_{\tau_{i+1}}-v_{\tau_i})\times\chi a
\cr\displaycause{because $\tau_i\le\sigma$, so
$a\Bsubseteq\Bvalue{\tau_i<\max\Cal S}$ and  $z_{\tau_i}\times\chi a=\chi a$}&
=S_I(\pmb z\times\pmb u,d\pmb v)\times\chi a;
\cr}$$

\noindent similarly,

$$\eqalignno{S_I(\pmb u,d\pmb v)\times\chi a
&=\sum_{i=0}^{n-1}u_{\tau_i}\times
  (v_{\tau_{i+1}}\times z_{\tau_{i+1}}-v_{\tau_i}\times z_{\tau_i})
  \times\chi a
\cr&=S_I(\pmb u,d(\pmb z\times\pmb v))\times\chi a.  \text{ \Qed}
\cr}$$

\noindent Taking the limit as $I\uparrow\Cal I(\Cal S\wedge\sigma)$,

\Centerline{$\chi a\times\int_{\Cal S\wedge\sigma}\pmb u\,d\pmb v
=\chi a\times\int_{\Cal S\wedge\sigma}\pmb z\times\pmb u\,d\pmb v
=\chi a\times\int_{\Cal S\wedge\sigma}\pmb u\,d(\pmb z\times\pmb v)$,}

\noindent while of course $\chi a=z_{\sigma}$.   As $\sigma$ is arbitrary,

\Centerline{$\pmb z\times\iivu
=\pmb z\times ii_{\pmb v}(\pmb z\times\pmb u)
=\pmb z\times ii_{\pmb z\times\pmb v}(\pmb u)$.}

\medskip

\quad{\bf (ii)}
Repeating part of the calculation when $\sigma=\max\Cal S$, we see that
if $\tau_0\le\tau_1$ in $\Cal S$, then

\Centerline{$\Bvalue{u_{\tau_0}\times(v_{\tau_1}-v_{\tau_0})\ne 0}
\Bsubseteq\Bvalue{\tau_0<\tau_1}
\Bsubseteq\Bvalue{\tau_0<\max\Cal S}
=\Bvalue{z_{\tau_0}=1}$,}

\noindent so, in the language of 613F,

$$\eqalign{\Delta_{c(\tau_0,\tau_1)}(\pmb u,d\pmb v)
&=u_{\tau_0}\times(v_{\tau_1}-v_{\tau_0})
\cr&=u_{\tau_0}\times z_{\tau_0}\times(v_{\tau_1}-v_{\tau_0})
=\Delta_{c(\tau_0,\tau_1)}(\pmb z\times\pmb u,d\pmb v).
\cr}$$

\noindent It follows immediately that
$S_I(\pmb u,d\pmb v)=S_I(\pmb z\times\pmb u,d\pmb v)$ for every
$I\in\Cal I(\Cal S)$ and
$\int_{\Cal S}\pmb u\,d\pmb v
=\int_{\Cal S}\pmb z\times\pmb u\,d\pmb v$.

\medskip

{\bf (b)} $\pmb z\times\pmb w$ is of bounded variation (614Q(a-ii)), so the
integral $\int_{\Cal S}|d(\pmb z\times\pmb w)|$ is defined.
Suppose that $\sigma_0\le\ldots\le\sigma_n=\max\Cal S$ and
$\sigma_i\in\Cal S$ for each $i\le n$.
Set $a_0=\Bvalue{\sigma_0=\max\Cal S}$
and $a_j=\Bvalue{\sigma_{j-1}<\sigma_j}\Bcap\Bvalue{\sigma_j=\max\Cal S}$
for
$1\le j\le n$.   Then $\langle a_j\rangle_{j\le n}$ is a partition of unity
in $\frak A$.   Set $y=\sum_{i=0}^{n-1}
|w_{\sigma_{i+1}}\times z_{\sigma_{i+1}}-w_{\sigma_i}\times z_{\sigma_i}|$.
Then

\Centerline{$a_0\Bsubseteq\inf_{i\le n}\Bvalue{z_{\sigma_i}=0}
\Bsubseteq\Bvalue{z=0}$}

\noindent and if $0\le j<n$

$$\eqalign{a_{j+1}
&\Bsubseteq\inf_{i<j}\Bvalue{w_{\sigma_{i+1}}\times z_{\sigma_{i+1}}
     -w_{\sigma_i}\times z_{\sigma_i}
    =w_{\sigma_{i+1}}-w_{\sigma_i}}
\cr&\mskip50mu
  \Bcap\Bvalue{w_{\sigma_{j+1}}\times z_{\sigma_{j+1}}
     -w_{\sigma_j}\times z_{\sigma_j}=-w_{\sigma_j}\times z_{\sigma_j}}
  \Bcap\Bvalue{z_{\sigma_j}=\chi 1}
\cr&\mskip50mu
  \Bcap\inf_{j<i<n}\Bvalue{w_{\sigma_{i+1}}\times z_{\sigma_{i+1}}
     -w_{\sigma_i}\times z_{\sigma_i}=0}
\cr&\Bsubseteq\Bvalue{y=\sum_{i=0}^{j-1}(|w_{\sigma_{i+1}}-w_{\sigma_i}|
  +|w_{\sigma_j}|}
\Bsubseteq\Bvalue{y\le w^{\uparrow}_{\sigma_j}+|w_{\sigma_j}|};
\cr}$$

\noindent since also $a_{j+1}\Bsubseteq\Bvalue{z_{\sigma_j}=1}$,

\Centerline{$a_{j+1}
\Bsubseteq\Bvalue{z\le(w^{\uparrow}_{\sigma_j}+|w_{\sigma_j}|)
  \times z_{\sigma_j}}
\Bsubseteq\Bvalue{z\le\sup|\pmb z\times(\pmb w^{\uparrow}+|\pmb w|)|}$.}

\noindent As $\sup_{j\le n}a_j=1$,
$y\le\sup|\pmb z\times(\pmb w^{\uparrow}+|\pmb w|)|$;
as $\sigma_0,\ldots,\sigma_n$
are arbitrary, $\int_{\Cal S}|d(\pmb z\times\pmb w)|
\le\sup|\pmb z\times(\pmb w^{\uparrow}+|\pmb w|)|$.

\medskip

{\bf (c)} Write $\bar w$ for $\sup|\pmb z\times\pmb w|$.
Suppose that $\sigma\in\Cal S$ and that
$\Cal I\in\Cal I(\Cal S\wedge\sigma)$ contains both $\min\Cal S$ and
$\sigma$.   Then, defining $\Osclln_I(\pmb w)$ as in 618B,
$|w_{\sigma}|\le\bar w+\Osclln_I(\pmb w)$.   \Prf\ Let
$(\tau_0,\ldots,\tau_n)$ linearly generate the $I$-cells, so that
$\tau_0=\min\Cal S$, $\tau_n=\sigma$ and
$\Osclln_I(\pmb w)=\sup_{i<n}|w_{\tau_{i+1}}-w_{\tau_i}|$.
Set $a_0=\Bvalue{\tau_0=\sigma}$ and
$a_i=\Bvalue{\tau_{i-1}<\sigma}\Bcap\Bvalue{\tau_i=\sigma}$ for
$1\le i\le n$, so that $\sup_{i\le n}a_i=1$.   For $1\le i\le n$,

$$\eqalign{a_i
&\Bsubseteq\Bvalue{w_{\sigma}=w_{\tau_i}}
   \Bcap\Bvalue{z_{\tau_{i-1}}=1}
\cr&\Bsubseteq\Bvalue{|w_{\sigma}|
\le|w_{\tau_i}-w_{\tau_{i-1}}|+|w_{\tau_{i-1}}\times z_{\tau_{i-1}}|}
\Bsubseteq\Bvalue{|w_{\sigma}|\le\Osclln_I(\pmb w)+\bar w};
\cr}$$

\noindent also, of course,

\Centerline{$a_0\Bsubseteq\Bvalue{w_{\sigma}=w_{\min\Cal S}}
\Bsubseteq\Bvalue{w_{\sigma}=0}
\Bsubseteq\Bvalue{|w_{\sigma}|\le\Osclln_I(\pmb w)+\bar w}$.}

\noindent As $\sup_{i\le n}a_i=1$,
$|w_{\sigma}|\le\Osclln_I(\pmb w)+\bar w$.\ \Qed

So if $I\in\Cal I(\Cal S)$ contains $\min\Cal S$ and $\sigma$,

\Centerline{$|w_{\sigma}|
\le\Osclln_{I\wedge\sigma}(\pmb w)+\bar w
\le\Osclln_I(\pmb w)+\bar w$;}

\noindent it follows that $|w_{\sigma}|\le\Osclln^*_I(\pmb w)+\bar w$ for
every $I\in\Cal S$, and taking the infimum over $I$ we have
$|w_{\sigma}|\le\Osclln(\pmb w)+\bar w$.   As $\sigma$ is arbitrary,
$\sup|\pmb w|\le\Osclln(\pmb w)+\bar w$, as claimed.

\medskip

{\bf (d)} If $\sigma\in\Cal S$ then

$$\eqalign{\Bvalue{\sigma<\max\Cal S}&
\Bsubseteq\Bvalue{|w_{\sigma}|\le\alpha}
\Bsubseteq\Bvalue{|w_{\sigma}\times z_{\sigma}|\le\alpha}\cr
\Bvalue{\max\Cal S\le\sigma}
&\Bsubseteq\Bvalue{w_{\sigma}=w_{\max\Cal S}}\Bcap\Bvalue{z_{\sigma}=0}
\Bsubseteq\Bvalue{|w_{\sigma}\times z_{\sigma}|\le\alpha},\cr}$$

\noindent So $|w_{\sigma}|\le\alpha\chi 1\vee|w_{\max\Cal S}|$
and $|w_{\sigma}\times z_{\sigma}|\le\alpha\chi 1$;  as $\sigma$
is arbitrary, $\pmb w$ is order-bounded,
$\sup|\pmb z\times\pmb w|\le\alpha\chi 1$ and
$\|\sup|\pmb z\times\pmb w|\|_{\infty}\le\alpha$.

Now (c) tells us that

\Centerline{
$\|\sup|\pmb w|\|_{\infty}
\le\|\sup|\pmb z\times\pmb w|\|_{\infty}+\|\Osclln(\pmb w)\|_{\infty}
\le\alpha+\|\Osclln(\pmb w)\|_{\infty}$.}

\medskip

{\bf (e)} Looking at the definition of $\pmb u_<$ in 641E-641F, we see that if
$\pmb u=\sfamilysS{u}$ and $\pmb z\times\pmb u=\sfamilysS{v}$
then whenever $\sigma\in I\in\Cal I(\Cal S)$
we have a disjoint family
$\langle a_{\sigma'}\rangle_{\sigma'\in I}$ such
that

$$\eqalign{a_{\sigma'}
&\Bsubseteq\Bvalue{\sigma'<\sigma}
   \Bcap\Bvalue{u_{I<\sigma}=u_{\sigma'}}
   \Bcap\Bvalue{v_{I<\sigma}=v_{\sigma'}}
\cr&\Bsubseteq\Bvalue{u_{\sigma'}=v_{\sigma'}}
   \Bcap\Bvalue{u_{I<\sigma}=u_{\sigma'}}
   \Bcap\Bvalue{v_{I<\sigma}=v_{\sigma'}}
\Bsubseteq\Bvalue{u_{I<\sigma}=v_{I<\sigma}}
\cr}$$

\noindent for every $\sigma'\in I$, while

\Centerline{$1\Bsetminus\sup_{\sigma'\in I}a_{\sigma'}
\Bsubseteq\Bvalue{u_{I<\sigma}=0}\Bcap\Bvalue{v_{I<\sigma}=0}
\Bsubseteq\Bvalue{u_{I<\sigma}=v_{I<\sigma}}$.}

\noindent So $u_{I<\sigma}=v_{I<\sigma}$;  as $I$ is arbitrary,
$u_{<\sigma}=v_{<\sigma}$;  as $\sigma$ is arbitrary,
$\pmb u_<=(\pmb z\times\pmb u)_<$.
}%end of proof of 654E

\leader{654G}{Lemma} Let $\Cal S$ be a sublattice of $\Cal T$.
Write $\Mmo^0$ for the space of moderately oscillatory processes
$\pmb u=\sfamilysS{u}$ with starting value $0$.
For an integrator $\pmb v$, write $\pmb v^*$ for its quadratic variation.
Suppose that $\pmb u\in\Mmo=\Mmo(\Cal S)$ and
that $\pmb w$, $\pmb w'\in\Mmo^0$ are such that
$\pmb w$ is a virtually local martingale and $\pmb w'$ is of bounded variation;
set $\pmb v=\pmb w+\pmb w'$.   Then

\Centerline{$\|\sup|\iivu|\|_2
\le 2(\sqrt{\|\pmb w^*\|_{\infty}}
  +\|\int_{\Cal S}|d\pmb w'|\|_{\infty})\|\sup|\pmb u|\|_2$.}

\proof{{\bf (a)} To begin with (down to the end of (c) below), suppose that
$\Cal S$ is full.   For a fully adapted process
$\pmb u$ with domain $\Cal S$, let $j(\pmb u)$ be the infimum of sums
$\sqrt{\|\sup|\pmb z^*|\|_1}+\|\int_{\Cal S}|d\pmb z'|\|_2$ where
$\pmb z$, $\pmb z'\in\Mmo^0$, $\pmb z$ is a virtually local martingale,
$\pmb z'$ is of bounded variation, and
$|\pmb u|\le|\pmb z|+|\pmb z'|$, counting $j(\pmb u)$ as $\infty$ if there are
no such $\pmb z$ and $\pmb z'$.   (By 623Kd, we shall be
able to speak of the quadratic variation $\pmb z^*$.)

We find that $\|\sup|\pmb u|\|_2\le 2j(\pmb u)$ for every
$\pmb u\in\Mmo$.   \Prf\ If $j(\pmb u)=\infty$, this is trivial.
Otherwise, let $\epsilon>0$.   Take
$\pmb z$, $\pmb z'\in\Mmo^0$ such that $\pmb z$ is a virtually local martingale,
$\pmb z'$ is of bounded variation,
$|\pmb u|\le|\pmb z|+|\pmb z'|$ and
$\sqrt{\|\sup|\pmb z^*|\|_1}+\|\int_{\Cal S}|d\pmb z'|\|_2
\le j(\pmb u)+\epsilon$.   Now, expressing $\pmb z$ as $\sfamilysS{z}$,

$$\eqalignno{\|\sup|\pmb z|\|_2&
\le 2\sup_{\sigma\in\Cal S}\|z_{\sigma}\|_2
\Displaycause{623M, because $\pmb z$ is an approximately local martingale, by
623J}
\le 2\sup_{\sigma\in\Cal S}\sqrt{\|z^*_{\sigma}\|_1}
\Displaycause{624H}
\le 2\sqrt{\|\sup|\pmb z^*|\|_1}.\cr}$$

\noindent On the other side,
$\sup|\pmb z'|\le\int_{\Cal S}|d\pmb z'|$ because $\pmb z'$ starts from $0$, so
$\|\sup|\pmb z'|\|_2\le\|\int_{\Cal S}|d\pmb z'|\|_2$.   Putting these
together,

$$\eqalign{\|\sup|\pmb u|\|_2
&\le\|\sup|\pmb z|+\sup|\pmb z'|\|_2
\le\|\sup|\pmb z|\|_2+\|\sup|\pmb z'|\|_2
\cr&\le 2\sqrt{\|\sup|\pmb z^*|\|_1}
  +\|\int_{\Cal S}|d\pmb z'|\|_2
\le 2(j(\pmb u)+\epsilon).\cr}$$

\noindent As $\epsilon$ is arbitrary, we have the result.\ \Qed

\medskip

{\bf (b)} Take $\pmb u$, $\pmb w$, $\pmb w'$ and $\pmb v$ as in the
statement of the lemma.   Then
$j(\iivu)\le\|\sup|\pmb u|\|_2
  (\sqrt{\|\pmb w^*\|_{\infty}}+\|\int_{\Cal S}|d\pmb w'|\|_{\infty})$.
\Prf\ If $\pmb u=\tbf 0$ then $\iivu=0$ and the result
is trivial.   If $\pmb u\ne\tbf 0$ and either
$\sup|\pmb w^*|$ or $\int_{\Cal S}|d\pmb w'|$ is not in $L^{\infty}$,
the result is again trivial.   So suppose that both
$\sup|\pmb w^*|$ and $\int_{\Cal S}|d\pmb w'|$ belong to
$L^{\infty}$.   Setting $\pmb y=ii_{\pmb w}(\pmb u)$
and $\pmb y'=ii_{\pmb w'}(\pmb u)$, $\iivu=\pmb y+\pmb y'$.
Now $\pmb y$ is a virtually local martingale (623O),
$\pmb y'$ is of bounded variation (614U), and both start from $0$.

Next,
$\pmb y^*=ii_{\pmb w^*}(\pmb u^2)$ (617Q(a-iii)), so

$$\eqalignno{\sqrt{\|\sup|\pmb y^*|\|_1}
&=\sqrt{\|\sup|ii_{\pmb w^*}(\pmb u^2)|\|_1}
=\sqrt{\|\int_{\Cal S}|d(ii_{\pmb w^*}(\pmb u^2))|\|_1}
\cr\displaycause{because $ii_{\pmb w^*}(\pmb u^2)$ is non-decreasing
and starts from $0$}&
\le\sqrt{\|\sup|\pmb u^2|\times\int_{\Cal S}d\pmb w^*\|_1}
\cr\displaycause{614U}&
\le\sqrt{\|\sup|\pmb u^2|\|_1\,\|\int_{\Cal S}d\pmb w^*\|_{\infty}}
=\sqrt{\|(\sup|\pmb u|)^2\|_1}
  \sqrt{\|\pmb w^*\|_{\infty}}
\Displaycause{because $\pmb w^*$ is non-decreasing and starts from $0$}
=\|\sup|\pmb u|\|_2\sqrt{\|\pmb w^*\|_{\infty}}.\cr}$$

\noindent On the other side,

\Centerline{$\int_{\Cal S}|d\pmb y'|
\le\sup|\pmb u|\times\int_{\Cal S}|d\pmb w'|$}

\noindent (614U again), so
$\|\int_{\Cal S}|d\pmb y'|\|_2
\le\|\sup|\pmb u|\|_2\|\int_{\Cal S}|d\pmb w'|\|_{\infty}$.   Putting
these together,

$$\eqalign{j(\iivu)
&\le\sqrt{\|\sup|\pmb y^*|\|_1}+\|\int_{\Cal S}|d\pmb y'|\|_2
\cr&\le\|\sup|\pmb u|\|_2
  \bigl(\sqrt{\|\pmb w^*\|_{\infty}}
    +\|\int_{\Cal S}|d\pmb w'|\|_{\infty}\bigr),
\cr}$$

\noindent as claimed.\ \Qed

\medskip

{\bf (c)} Combining this with (a), we see that

\Centerline{$\|\sup|\iivu|\|_2
\le 2j(\iivu)
\le 2(\sqrt{\|\pmb w^*\|_{\infty}}
  +\|\int_{\Cal S}|d\pmb w'|\|_{\infty})\|\sup|\pmb u|\|_2$.}

\medskip

{\bf (d)} This proves the result when $\Cal S$ is full.   For the general case,
let $\hat\Cal S$ be the covered envelope of $\Cal S$ and $\hatpmb u$,
$\hatpmb w$, $\hatpmb w'$, $\hatpmb v$, $\hatpmb w^*$ 
the fully adapted extensions of
$\pmb u$, $\pmb w$, $\pmb w'$, $\pmb v$, $\pmb w^*$ to $\hat\Cal S$.   Then
$\hatpmb u$, $\hatpmb w$ and $\hatpmb w'$ belong to $\Mmo(\hat\Cal S)$ 
(615F(a-vi)), while $\hatpmb w$ and $\hatpmb w'$ have starting value $0$
(615H), $\hatpmb w'$ is of bounded variation (614Q(a-iv-$\beta$)),
$\hatpmb v=\hatpmb w+\hatpmb w'$ and $\hatpmb w^*$ is the quadratic
variation of $\hatpmb w$ (617N).   By (a)-(c) above,

\Centerline{$\|\sup|ii_{\hatpmb v}(\hatpmb u)|\|_2
\le 2(\sqrt{\|\hatpmb w^*\|_{\infty}}
  +\|\int_{\hat\Cal S}|d\hatpmb w'|\|_{\infty})\|\sup|\hatpmb u|\|_2$.}

\noindent Now we know that $ii_{\hatpmb v}(\hatpmb u)$ is the fully adapted
extension of $\iivu$ (616Q(c-ii)), so that 
$\sup|ii_{\hatpmb v}(\hatpmb u)|=\sup|\iivu|$ (614Ga);  at the same time,

\Centerline{$\|\hatpmb w^*\|_{\infty}=\|\sup|\hatpmb w^*|\|_{\infty}
=\|\sup|\pmb w^*|\|_{\infty}=\|\pmb w^*\|_{\infty}$}

\noindent and $\sup|\hatpmb u|=\sup|\pmb u|$.   Finally, 
$\int_{\hat\Cal S}|d\hatpmb w'|=\int_{\Cal S}|d\pmb w'|$ by 
614Q(a-iv-$\beta$).   So

\Centerline{$\|\sup|ii_{\pmb v}(\pmb u)|\|_2
\le 2(\sqrt{\|\pmb w^*\|_{\infty}}
  +\|\int_{\Cal S}|d\pmb w'|\|_{\infty})\|\sup|\pmb u|\|_2$,}

\noindent as required.


}%end of proof of 654G

\leader{654H}{Lemma}\cmmnt{ (The key.)}
Let $\Cal S$ be a sublattice of $\Cal T$ with a greatest element.
Suppose that $h:\BbbR^2\to\Bbb R$ is
a continuous function and that $K\ge 0$ is such that
$|h(\alpha,\beta)-h(\alpha',\beta)|\le K|\alpha-\alpha'|$ for all $\alpha$,
$\alpha'$, $\beta\in\Bbb R$;  let $\pmb w=\sfamilysS{w}$,
$\pmb w'=\sfamilysS{w'}$ be processes with domain $\Cal S$
such that $\pmb w$ is a virtually local martingale,
$\pmb w'$ is of bounded variation and both start fom $0$.
Write $\pmb w^*$ for the quadratic variation of $\pmb w$,
$\pmb w^{\prime\uparrow}$ for the cumulative variation of
$\pmb w'$, and $\pmb z$ for
$\family{\sigma}{\Cal S}{\chi\Bvalue{\sigma<\max\Cal S}}$.   Suppose that

\Centerline{$2K(\sqrt{\|\pmb w^*\|_{\infty}}
   +2\|\pmb z\times\pmb w^{\prime\uparrow}\|_{\infty})<1$.\cmmnt{ \hfil(*)}}

\noindent Set $\pmb v=\pmb w+\pmb w'$.
Then for any $\pmb u_{\star}$, $\pmb y\in\Mmo=\Mmo(\Cal S)$
there is a unique $\pmb u\in\Mmo$ such that

\Centerline{$\pmb u=\pmb u_{\star}+ii_{\pmb v}(\bar h(\pmb u,\pmb y))$.}

\proof{ The proof will proceed in three steps, the first two being with
a variation on the hypothesis (*).

\medskip

{\bf (a)} Suppose that $\|\pmb u_{\star}\|_{\infty}$ and
$\|\pmb y\|_{\infty}$ are
both finite, and that instead of the hypothesis (*) we suppose that

\Centerline{$2K\gamma<1$ where $\gamma=
\sqrt{\|\pmb w^*\|_{\infty}}+\|\int_{\Cal S}|d\pmb w'|\|_{\infty}$.
\hfil(**)}

\medskip

\quad{\bf (i)} Define a sequence $\langle\pmb u_n\rangle_{n\ge 1}$
of processes with domain $\Cal S$ by saying that
$\pmb u_0=\pmb u_{\star}$ and

\Centerline{$\pmb u_{n+1}
=\pmb u_{\star}+ii_{\pmb v}(\bar h(\pmb u_n,\pmb y))$}

\noindent for every $n\in\Bbb N$.
Then every $\pmb u_n$ belongs to $\Mmo$
(induce on $n$, using 619Gc and 616J in the inductive step).   Set
$\pmb z_n=\pmb u_{n+1}-\pmb u_n$ for $n\ge 0$.   Then

$$\eqalign{\pmb z_{n+1}
&=\pmb u_{n+2}-\pmb u_{n+1}
=ii_{\pmb v}(\bar h(\pmb u_{n+1},\pmb y))
  -ii_{\pmb v}(\bar h(\pmb u_n,\pmb y))
\cr&
=ii_{\pmb v}(\bar h(\pmb u_{n+1},\pmb y)-\bar h(\pmb u_n,\pmb y))
\cr}$$

\noindent for every $n$.  Now 654G tells us that

$$\eqalignno{\|\sup|\pmb z_{n+1}|\|_2
&\le 2\gamma\|\sup|\bar h(\pmb u_{n+1},\pmb y)
   -\bar h(\pmb u_n,\pmb y)|\|_2
\le 2K\gamma\|\sup|\pmb u_{n+1}-\pmb u_n|\|_2
\cr\displaycause{by the Lipschitz condition on $h$}&
=2K\gamma\|\sup|\pmb z_n|\|_2\cr}$$

\noindent for $n\in\Bbb N$.   At the beginning of the iteration,

$$\eqalignno{\|\sup|\pmb z_0|\|_2
&=\|\sup|\pmb u_1-\pmb u_{\star}|\|_2
=\|\sup|ii_{\pmb v}(\bar h(\pmb u_{\star},\pmb y))|\|_2\cr&
\le 2\gamma
  \|\sup|\bar h(\pmb u_{\star},\pmb y)|\|_2
\le 2\gamma\|\bar h(\pmb u_{\star},\pmb y)\|_{\infty}\cr}$$

\noindent is finite because both
$\|\pmb u_{\star}\|_{\infty}$ and $\|\pmb y\|_{\infty}$ are
finite and $h$ is locally bounded.
As $2K\gamma<1$, $\sum_{n=0}^{\infty}\|\sup|\pmb z_n|\|_2$ is finite.

\medskip

\quad{\bf (ii)} Of course

\Centerline{$\theta(u)=\Expn(|u|\wedge\chi 1)
\le\|u\|_1\le\|u\|_2$}

\noindent for every $u\in L^0$.
So $\sum_{n=0}^{\infty}\theta(\sup|\pmb z_n|)<\infty$
and $\sequencen{\pmb u_n}$ is Cauchy for the ucp uniformity.
As $\Mmo$ is complete under the ucp uniformity (615F(a-iv)),
$\pmb u=\lim_{n\to\infty}\pmb u_n$ is defined in $\Mmo$.
Because $h$ is continuous,
$\bar h(\pmb u,\pmb y)=\lim_{n\to\infty}\bar h(\pmb u_n,\pmb y)$
(619H).   By 616J, applied to the integrating interval function
$\Delta\pmb v$ (616Ic),

\Centerline{$\pmb u_{\star}+ii_{\pmb v}(\bar h(\pmb u,\pmb y))
=\lim_{n\to\infty}\pmb u_{\star}+ii_{\pmb v}(\bar h(\pmb u_n,\pmb y))
=\lim_{n\to\infty}\pmb u_{n+1}
=\pmb u$.}

\medskip

\quad{\bf (iii)}
Thus we have an appropriate $\pmb u$.   It will be helpful to know that
$\sup|\pmb u|\in L^2$.
\Prf\  Since $\sum_{n=0}^{\infty}\|\sup|\pmb z_n|\|_2<\infty$,
$z=\sum_{n=0}^{\infty}\sup|\pmb z_n|$ is defined in
$L^2$;  now
$|\pmb u_{\star}|+z\ge\sup|\pmb u_n|$ for every $n\in\Bbb N$,
so $\sup|\pmb u|\le|\pmb u_{\star}|+z$ is square-integrable.\ \Qed

\medskip

{\bf (b)} When we come to prove that our solutions are unique, the
following formulation will be useful.   As in (a), assume that (**) is true.
Suppose that

\Centerline{$\pmb u_{1\star}$, $\pmb u_{1\star}'$,
$\pmb y'$, $\pmb y''$, $\pmb u_1$, $\pmb u_1'\in\Mmo$,}

\Centerline{$\pmb u_1=\pmb u_{1\star}+ii_{\pmb v}(\bar h(\pmb u_1,\pmb y'))$,
\quad$\pmb u_1'=\pmb u_{1\star}'+ii_{\pmb v}(\bar h(\pmb u_1',\pmb y''))$;}

\noindent suppose moreover that $\sup|\pmb u_1|\in L^2$, and set
$a=\Bvalue{\pmb u_{1\star}\ne\pmb u_{1\star}'}\Bcup\Bvalue{\pmb y'\ne\pmb y''}$.
Then $\Bvalue{\pmb u_1\ne\pmb u_1'}\Bsubseteq a$.
\Prf\ Let $\epsilon>0$.
Let $M\ge 0$ be such that $\bar\mu\Bvalue{\sup|\pmb u_1'|>M}\le\epsilon$
and set $a'=a\Bcup\Bvalue{\sup|\pmb u_1'|>M}$.
Set $\tildepmb u_0
=\med(-M\,\tbf 1^{(\Cal S)},\pmb u_1',M\,\tbf 1^{(\Cal S)})$ and
$\tildepmb u_{n+1}
=\pmb u_{1\star}+ii_{\pmb v}(\bar h(\tildepmb u_n,\pmb y'))$
for $n\in\Bbb N$.   Then

$$\eqalign{\|\sup|\tildepmb u_{n+1}-\pmb u_1|\|_2
&=\|\sup|ii_{\pmb v}
   (\bar h(\tildepmb u_n,\pmb y')-\bar h(\pmb u_1,\pmb y'))|\|_2
\cr&\le 2\gamma
  \|\sup|\bar h(\tildepmb u_n,\pmb y')-\bar h(\pmb u_1,\pmb y')|\|_2
\le 2K\gamma\|\sup|\tildepmb u_n-\pmb u_1|\|_2\cr}$$

\noindent for each $n$, while
$\|\sup|\tildepmb u_0-\pmb u_1|\|_2\le M+\|\sup|\pmb u_1|\|_2$
is finite.   So
$\lim_{n\to\infty}\|\sup|\tildepmb u_n-\pmb u_1|\|_2=0$ and
$\pmb u_1'=\lim_{n\to\infty}\tildepmb u_n$ for the ucp topology.

At the same time, we find that
$\Bvalue{\tildepmb u_n\ne\pmb u_1'}\Bsubseteq a'$ for every $n$.
To see this, induce on $n$.   At the start,
$\Bvalue{\tildepmb u_0\ne\pmb u_1'}
=\Bvalue{\sup|\pmb u_1'|>M}\Bsubseteq a'$.   For the inductive step,

$$\eqalign{\Bvalue{\tildepmb u_{n+1}\ne\pmb u_1'}
&=\Bvalue{\pmb u_{1\star}+ii_{\pmb v}(\bar h(\tildepmb u_n,\pmb y'))
  \ne\pmb u_{1\star}'+ii_{\pmb v}(\bar h(\pmb u_1',\pmb y''))}
\cr&\Bsubseteq\Bvalue{\pmb u_{1\star}\ne\pmb u_{1\star}'}
  \Bcup\Bvalue{\bar h(\tildepmb u_n,\pmb y')
  \ne\bar h(\pmb u_1',\pmb y'')}
\Bsubseteq a\Bcup\Bvalue{\tildepmb u_n\ne\pmb u_1'}
\Bsubseteq a',
\cr}$$

\noindent so the induction proceeds.

It follows that $\Bvalue{\pmb u_1\ne\pmb u_1'}\Bsetminus a$ has
measure at most $\epsilon$.   As $\epsilon$ is arbitrary,
$\Bvalue{\pmb u_1\ne\pmb u_1'}\Bsubseteq a$.\ \Qed

\medskip

{\bf (c)} For the second stage of the existence proof,
continue to assume (**),
but drop the $\|\,\|_{\infty}$-boundedness conditions on $\pmb u_{\star}$ and
$\pmb y$, and suppose only that $\pmb u_{\star}$, $\pmb y\in\Mmo$.

\medskip

\quad{\bf (i)} In this case, set

\Centerline{$\pmb u_{\star k}
=\med(-k\tbf 1^{(\Cal S)},\pmb u_{\star},k\tbf 1^{(\Cal S)})$,
\quad$\pmb y_k
=\med(-k\,\tbf 1^{(\Cal S)},\pmb y,k\,\tbf 1^{(\Cal S)})$,}

\Centerline{$a_k
=\Bvalue{\sup|\pmb u_{\star}|\ge k}\Bcup\Bvalue{\sup|\pmb y|\ge k}$}

\noindent for $k\in\Bbb N$.   By (a), there is for each
$k$ a $\pmb u_k\in\Mmo$ such that
$\pmb u_k
=\pmb u_{\star k}+ii_{\pmb v}(\bar h(\pmb u_k,\pmb y_k))$
and $\|\sup|\pmb u_k|\|_2<\infty$.   By (b),
$\Bvalue{\pmb u_k\ne\pmb u_l}\Bsubseteq a_k$ for
$l\ge k$.   Since $\sequence{k}{a_k}$ is a non-increasing sequence with
infimum $0$, $z=\sup_{k\in\Bbb N}\sup|\pmb u_k|
=\sup_{k\in\Bbb N}\chi(a_k\Bsetminus\chi a_{k+1})\times\sup|\pmb u_k|$
is defined in $L^0$ and
$\sup|\pmb u-\pmb u_k|\le z\times\chi a_k$ for each $k$.   This shows
that $\pmb u=\lim_{k\to\infty}\pmb u_k$ for the ucp topology, so
$\pmb u\in\Mmo$ and

\Centerline{$\pmb u
=\lim_{k\to\infty}\pmb u_{\star k}
    +ii_{\pmb v}(\bar h(\pmb u_k,\pmb y_k))
=\pmb u_{\star}+ii_{\pmb v}(\bar h(\pmb u,\pmb y))$}

\noindent as in (a-ii), but this time noting that
$\bar h(\pmb u,\pmb y)=\lim_{k\to\infty}\bar h(\pmb u_k,\pmb y_k)$
for the ucp topology.

\medskip

\quad{\bf (ii)} To see that $\pmb u$ is unique, let $\pmb u_1\in\Mmo$ be
such that $\pmb u_1=\pmb u_{\star}+ii_{\pmb v}(\bar h(\pmb u_1,\pmb y))$.
This time, (b) tells us that
$\Bvalue{\pmb u_1\ne\pmb u_k}\Bsubseteq a_k$ for each $k$.
So $\pmb u_1=\pmb u$.

\medskip

{\bf (d)} We are now ready to tackle the result under the given hypothesis
(*).

\medskip

\quad{\bf (i)} Setting $\pmb w_1'=\pmb z\times\pmb w'$, we see from
614Q(a-ii), or otherwise, that $\pmb w_1'$ is of bounded variation, and from
654Eb that

$$\eqalign{\|\int_{\Cal S}|d\pmb w_1'|\|_{\infty}
&\le\|\sup|\pmb z\times(\pmb w^{\prime\uparrow}+|\pmb w'|)|\|_{\infty}
\cr&=\|\pmb z\times(\pmb w^{\prime\uparrow}+|\pmb w'|)\|_{\infty}
\le 2\|\pmb z\times\pmb w^{\prime\uparrow}\|_{\infty}
\cr}$$

\noindent because $\pmb w'$ starts from $0$.   We therefore have

\Centerline{$2K(\sqrt{\|\pmb w^*\|_{\infty}}
  +\|\int_{\Cal S}|d\pmb w_1'|\|_{\infty})
<1$,}

\noindent and if we set $\pmb v_1=\pmb w+\pmb w_1'$, (c) tells us that
there is a near-simple process $\pmb u_1$ such that
$\pmb u_1=\pmb u_{\star}+ii_{\pmb v_1}(\bar h(\pmb u_1,\pmb y))$.   Set
$\pmb w_2'=\pmb w'-\pmb w_1'$ and
$\pmb u=\pmb u_1+ii_{\pmb w_2'}(\bar h(\pmb u_1,\pmb y))$.

We see that

\Centerline{$\pmb z\times\pmb v_1
=\pmb z\times\pmb w+\pmb z\times\pmb w_1'
=\pmb z\times\pmb w+\pmb z\times\pmb z\times\pmb w'
=\pmb z\times\pmb w+\pmb z\times\pmb w'=\pmb z\times\pmb v$,}

$$\eqalignno{\pmb z\times\pmb u
&=\pmb z\times\pmb u_1
   +\pmb z\times ii_{\pmb w_2'}(\bar h(\pmb u_1,\pmb y))
=\pmb z\times\pmb u_1
   +\pmb z\times ii_{\pmb z\times\pmb w_2'}(\bar h(\pmb u_1,\pmb y))
\cr\displaycause{654Ea}&
=\pmb z\times\pmb u_1
\cr}$$

\noindent because $\pmb z\times\pmb w_2'=0$.   Note also that

$$\eqalignno{\pmb z\times\bar h(\pmb z\times\pmb u,\pmb y)
&=\pmb z\times\bar h(\pmb z\times\pmb z\times\pmb u,\pmb z\times\pmb y)
\cr\displaycause{619Ge}&
=\pmb z\times\bar h(\pmb z\times\pmb u,\pmb z\times\pmb y)
=\pmb z\times\bar h(\pmb u,\pmb y).
\cr}$$

\noindent Consequently

$$\eqalignno{\pmb z\times ii_{\pmb v}(\bar h(\pmb u,\pmb y))
&=\pmb z\times ii_{\pmb z\times\pmb v}(\pmb z\times\bar h(\pmb u,\pmb y))
\cr\displaycause{654Ea again}&
=\pmb z\times ii_{\pmb z\times\pmb v}
  (\pmb z\times\bar h(\pmb z\times\pmb u,\pmb y))
=\pmb z\times ii_{\pmb z\times\pmb v_1}
  (\pmb z\times\bar h(\pmb z\times\pmb u_1,\pmb y))
\cr&=\pmb z\times ii_{\pmb v_1}(\bar h(\pmb u_1,\pmb y))
=\pmb z\times(\pmb u_1-\pmb u_{\star})
=\pmb z\times(\pmb u-\pmb u_{\star}),
\cr}$$

\noindent that is,

\Centerline{$\pmb z\times
   (\pmb u-\pmb u_{\star}-ii_{\pmb v}(\bar h(\pmb u,\pmb y)))=0$.}

\noindent Expressing $\pmb u$ and $\pmb u_{\star}$ as $\sfamilysS{u}$
and $\family{\sigma}{\Cal S}{u_{\star\sigma}}$, this means that for any
$\sigma\in\Cal S$ we have

\Centerline{$\Bvalue{\sigma<\max\Cal S}
\Bsubseteq\Bvalue{u_{\sigma}
  =u_{\star\sigma}
     +\int_{\Cal S\wedge\sigma}\bar h(\pmb u,\pmb y)d\pmb v}$.}

\noindent At the top end, we can calculate

$$\eqalignno{u_{\max\Cal S}
&=u_{\star,\max\Cal S}+\int_{\Cal S}\bar h(\pmb u_1,\pmb y)d\pmb v_1
   +\int_{\Cal S}\bar h(\pmb u_1,\pmb y)d\pmb w_2'
\cr&=u_{\star,\max\Cal S}+\int_{\Cal S}\bar h(\pmb u_1,\pmb y)d\pmb v
=u_{\star,\max\Cal S}+\int_{\Cal S}\pmb z\times\bar h(\pmb u_1,\pmb y)d\pmb v
\cr\displaycause{by the other part of 654Ea}&
=u_{\star,\max\Cal S}
+\int_{\Cal S}\pmb z\times\bar h(\pmb z\times\pmb u_1,\pmb y)d\pmb v
\cr\noalign{\noindent and, unwinding,}&
=u_{\star,\max\Cal S}
  +\int_{\Cal S}\pmb z\times\bar h(\pmb z\times\pmb u,\pmb y)d\pmb v
\cr&=u_{\star,\max\Cal S}+\int_{\Cal S}\pmb z\times\bar h(\pmb u,\pmb y)d\pmb v
=u_{\star,\max\Cal S}+\int_{\Cal S}\bar h(\pmb u,\pmb y)d\pmb v.
\cr}$$

But this means that, for any $\sigma\in\Cal S$,

\Centerline{$\Bvalue{\sigma=\max\Cal S}
\Bsubseteq\Bvalue{u_{\sigma}=u_{\star\sigma}
  +\int_{\Cal S\wedge\sigma}\bar h(\pmb u,\pmb y)d\pmb v}$}

\noindent and
$u_{\sigma}=u_{\star\sigma}+\int_{\Cal S\wedge\sigma}\bar h(\pmb u,d\pmb y)d\pmb v$.
As $\sigma$ is arbitrary,
$\pmb u=\pmb u_{\star}+ii_{\pmb v}(\bar h(\pmb u,d\pmb y))$.

\medskip

\quad{\bf (ii)}
To show uniqueness we can run the argument backwards, as follows.
Suppose that $\tildepmb u$ is a near-simple process such that
$\tildepmb u=\pmb u_{\star}+ii_{\pmb v}(\bar h(\tildepmb u,\pmb y))$.
Set $\tildepmb u'=\pmb u_{\star}+ii_{\pmb v_1}(\bar h(\tildepmb u,\pmb y))$.
Then

$$\eqalign{\pmb z\times\tildepmb u'
&=\pmb z\times\pmb u_{\star}
  +\pmb z\times ii_{\pmb z\times\pmb v_1}(\bar h(\tildepmb u,\pmb y))
\cr&=\pmb z\times\pmb u_{\star}
  +\pmb z\times ii_{\pmb z\times\pmb v}(\bar h(\tildepmb u,\pmb y))
=\pmb z\times\tildepmb u
\cr}$$

\noindent so that

$$\eqalign{\pmb z\times\tildepmb u'
&=\pmb z\times\pmb u_{\star}+\pmb z\times ii_{\pmb v_1}(\bar h(\tildepmb u,\pmb y))
=\pmb z\times\pmb u_{\star}
  +\pmb z\times ii_{\pmb v_1}(\bar h(\pmb z\times\tildepmb u,\pmb y))
\cr&=\pmb z\times\pmb u_{\star}
  +\pmb z\times ii_{\pmb v_1}(\bar h(\pmb z\times\tildepmb u',\pmb y))
=\pmb z\times\pmb u_{\star}
  +\pmb z\times ii_{\pmb v_1}(\bar h(\tildepmb u',\pmb y)),
\cr}$$

\noindent while

$$\eqalign{u_{\star,\max\Cal S}
  +\int_{\Cal S}\bar h(\tildepmb u,\pmb y)d\pmb v_1
&=u_{\star,\max\Cal S}
  +\int_{\Cal S}\pmb z\times\bar h(\tildepmb u,\pmb y)d\pmb v_1
\cr&=u_{\star,\max\Cal S}
   +\int_{\Cal S}\pmb z\times\bar h(\tildepmb u',\pmb y)d\pmb v_1
=u_{\star,\max\Cal S}+\int_{\Cal S}\bar h(\tildepmb u',\pmb y)d\pmb v_1.
\cr}$$

\noindent As in (i) just above, this is enough to show that

\Centerline{$\tildepmb u'
=\pmb u_{\star}+ii_{\pmb v_1}(\bar h(\tildepmb u',\pmb y))$,}

\noindent so that $\tildepmb u'=\pmb u_1$, because we know from (c-ii)
that we have a unique solution to the equation defining $\pmb u_1$.
Consequently

\Centerline{$\pmb z\times\tildepmb u
=\pmb z\times\tildepmb u'
   +\pmb z\times ii_{\pmb w_2'}(\bar h(\tildepmb u,d\pmb y))
=\pmb z\times\pmb u_1
=\pmb z\times\pmb u$;}

\noindent and once again

$$\eqalign{u_{\star,\max\Cal S}
  +\int_{\Cal S}\bar h(\tildepmb u,\pmb y)d\pmb v
&=u_{\star,\max\Cal S}
  +\int_{\Cal S}\bar h(\pmb z\times\tildepmb u,\pmb y)d\pmb v
\cr&=u_{\star,\max\Cal S}
  +\int_{\Cal S}\bar h(\pmb z\times\pmb u,\pmb y)d\pmb v
=u_{\star,\max\Cal S}+\int_{\Cal S}\bar h(\pmb u,\pmb y)d\pmb v,
\cr}$$

\noindent so $\tildepmb u$ and $\pmb u$ agree at $\max\Cal S$ and are
equal.   Thus the solution assembled in (i) is unique.

This completes the proof under the hypothesis (*).
}%end of proof of 654H

\leader{654I}{Theorem} Suppose that $\family{t}{T}{\frak A_t}$ is
right-continuous.   Let $\Cal S$ be an order-convex sublattice of
$\Cal T$ with a least member.   Suppose that
$h:\BbbR^2\to\Bbb R$ is
a continuous function and that $K\ge 0$ is such that
$|h(\alpha,\beta)-h(\alpha',\beta)|\le K|\alpha-\alpha'|$ for all $\alpha$,
$\alpha'$, $\beta\in\Bbb R$.   Let $\pmb v$ be a locally near-simple
local integrator
with domain $\Cal S$.   Then for any locally moderately oscillatory processes
$\pmb u_{\star}$, $\pmb y$ with domain $\Cal S$ there is a unique
locally moderately oscillatory process $\pmb u$ with domain $\Cal S$ such that

\Centerline{$\pmb u=\pmb u_{\star}+ii_{\pmb v}(\bar h(\pmb u,\pmb y))$.}

\proof{{\bf (a)} Let $\epsilon>0$ be such that
$2K(\sqrt{\epsilon}+3\epsilon)<1$.   By the Bichteler-Dellacherie theorem
(627J), $\pmb v$ is a \sm;  by the Fundamental Theorem of Martingales
(643M), we can express it as $\pmb w+\pmb w'$ where
$\pmb w=\sfamilysS{w}$ is a local martingale,
$\pmb w'=\sfamilysS{w'}$ is locally of bounded variation and
$\Osclln(\pmb w\restr\Cal S\wedge\tau)\le\epsilon\chi 1$ for every
$\tau\in\Cal S$.

\medskip

{\bf (b)} For the time being (down to the end of (f) below),
suppose that $\Cal S$ has a greatest member and that $\pmb w$ is a
martingale.   Then
$\pmb u_{\star}$ and $\pmb y$ moderately oscillatory, $\pmb v$ is
near-simple and $\Osclln(\pmb w)\le\epsilon\chi 1$.
Express $\pmb u_{\star}$ as
$\family{\sigma}{\Cal S}{u_{\star\sigma}}$.

We know that $\pmb w$ is
near-simple (632Ia), so $\pmb w'$ also is near-simple,
and its cumulative
variation $\pmb w^{\prime\uparrow}=\sfamilysS{w^{\prime\uparrow}}$ is
near-simple (631K).
Let $\pmb w^*=\sfamilysS{w^*}$ be the quadratic variation of
$\pmb w$;  this too is near-simple (631Ja).

\medskip

{\bf (c)} By 631Ra, there is a non-decreasing
sequence $\sequencen{\tau_n}$ in $\Cal S$ such that $\tau_0=\min\Cal S$,
$\inf_{n\in\Bbb N}\Bvalue{\tau_n<\max\Cal S}=0$ and whenever $n\in\Bbb N$
and $\sigma\in[\tau_n,\tau_{n+1}]$,

\Centerline{$\Bvalue{|w^{\prime\uparrow}_{\sigma}+w^*_{\sigma}
   -w^{\prime\uparrow}_{\tau_n}-w^*_{\tau_n}|
  \ge\epsilon}
\Bsubseteq\Bvalue{\sigma=\tau_{n+1}}$.}

\medskip

{\bf (d)} For $n\in\Bbb N$, write $\Cal S_n=\Cal S\wedge\tau_n$.   Then we
have a sequence $\sequencen{\pmb u_n}$ of moderately oscillatory processes
such that

\Centerline{$\pmb u_n=\pmb u_{\star}\restr\Cal S_n
  +ii_{\pmb v\restr\Cal S_n}(\bar h(\pmb u_n,\pmb y\restr\Cal S_n))$}

\noindent for each $n$.   \Prf\ Induce on $n$.   The induction starts with
$\pmb u_0=\pmb u_{\star}\restr\Cal S_0$ where
$\Cal S_0=\{\min\Cal S\}$.   For the inductive step to $n+1$, given
$\pmb u_n=\family{\sigma}{\Cal S_n}{u_{n\sigma}}$, set

\Centerline{$\tilde{\Cal S}_n=[\tau_n,\tau_{n+1}]$,
\quad$\tildepmb u_{\star n}=\pmb u_{\star}\restr\tilde{\Cal S}_n
  +(u_{n\tau_n}-u_{\star\tau_n})\tbf 1^{(\tilde{\Cal S}_n)}$,}

\Centerline{$\tildepmb w
  =\pmb w\restr\tilde{\Cal S}_n-w_{\tau_n}\tbf 1^{(\tilde{\Cal S}_n)}$,
\quad$\tildepmb w'
=\pmb w'\restr\tilde{\Cal S}_n-w'_{\tau_n}\tbf 1^{(\tilde{\Cal S}_n)}$,
\quad$\tildepmb v=\tildepmb w+\tildepmb w'$.}

\noindent Then $\tildepmb w$ and $\tildepmb w'$ are near-simple
processes with domain $\tilde{\Cal S}_n$ (631Fa), $\tildepmb w$ is a
martingale (622Db), $\tildepmb w'$ is of bounded variation (614Lc)
and both start with value $0$ at $\min\tilde{\Cal S}_n=\tau_n$.
The quadratic variation $\tildepmb w^*
=\family{\sigma}{\tilde{\Cal S}_n}{\tilde w^*_{\sigma}}$
of $\tildepmb w$ is
$\pmb w^*\restr\tilde{\Cal S}_n-w^*_{\tau_n}\tbf 1^{(\tilde{\Cal S}_n)}$
(use both halves of 617Kb), and the cumulative variation
$\tildepmb w^{\prime\uparrow}
=\family{\sigma}{\tilde{\Cal S}_n}{\tilde w^{\prime\uparrow}_{\sigma}}$
of $\tildepmb w'$ is
$\pmb w^{\prime\uparrow}\restr\tilde{\Cal S}_n
  -w^{\prime\uparrow}_{\tau_n}\tbf 1^{(\tilde{\Cal S}_n)}$
(614Pb).

We know that, for $\sigma\in\tilde{\Cal S}_n$,

\Centerline{$\Bvalue{\sigma<\tau_{n+1}}
\Bsubseteq\Bvalue{|w^{\prime\uparrow}_{\sigma}-w^{\prime\uparrow}_{\tau_n}|<\epsilon}
\Bsubseteq\Bvalue{|\tilde w^{\prime\uparrow}_{\sigma}|\le\epsilon}$,}

\noindent so 654Ed tells us that
$\|\sup|\pmb z\times\tildepmb w^{\prime\uparrow}|\|_{\infty}\le\epsilon$, where
$\pmb z=\family{\sigma}{\tilde{\Cal S}_n}{\chi\Bvalue{\sigma<\tau_{n+1}}}$.
Similarly,

\Centerline{$\Bvalue{\sigma<\tau_{n+1}}
\Bsubseteq\Bvalue{|w^*_{\sigma}-w^*_{\tau_n}|<\epsilon}
\Bsubseteq\Bvalue{|\tilde w^*_{\sigma}|\le\epsilon}$,}

\noindent and by the other half of 654Ed,

$$\eqalignno{\|\tildepmb w^*\|_{\infty}
&\le\epsilon+\|\Osclln(\tildepmb w^*)\|_{\infty}
=\epsilon+\|(\Osclln(\tildepmb w)^2\|_{\infty}
\Displaycause{618Sb}
=\epsilon+\|\Osclln(\tildepmb w)\|^2_{\infty}
=\epsilon+\|\Osclln(\pmb w\restr\tilde{\Cal S}_n)\|^2_{\infty}
\le\epsilon+\|\Osclln(\pmb w)\|^2_{\infty}
\le\epsilon+\epsilon^2.
\cr}$$

\noindent Accordingly

\Centerline{$2K\bigl(\sqrt{\|\tildepmb w^*\|_{\infty}}
  +2\|\pmb z\times\tildepmb w^{\prime\uparrow}\|_{\infty}\bigr)
\le 2K(\sqrt{\epsilon+\epsilon^2}+2\epsilon)
\le 2K(\sqrt{\epsilon}+3\epsilon)
<1$}

\noindent by the choice of $\epsilon$.   We can therefore apply 654H
to see
that there is a moderately oscillatory process $\tildepmb u_n$ with domain
$\tilde{\Cal S}_n$ such that
$\tildepmb u_n=\tildepmb u_{\star n}
  +ii_{\tildepmb v}(\bar h(\tildepmb u_n,\tildepmb y))$.

Since the processes
$\pmb u_n$ and $\tildepmb u_n$ take the same value at $\tau_n$, they have
a common extension $\pmb u_n\cup\tildepmb u_n$
to $\Cal S_n\cup\tilde{\Cal S}_n$, which is a covering
sublattice of $\Cal S_{n+1}$ (if
$\sigma\in\Cal S_{n+1}=[\tau_0,\tau_{n+1}]$ then $\{\sigma\}$ is covered,
in the sense of 611M, by
$\{\sigma\wedge\tau_n,\sigma\vee\tau_n\}
\subseteq\Cal S_n\cup\tilde{\Cal S}_n$).
We therefore have a fully adapted process $\pmb u_{n+1}$ with domain
$\Cal S_{n+1}$ extending $\pmb u_n\cup\tildepmb u_n$.
By 615F(a-v), $\pmb u_n\cup\tildepmb u_n$ is moderately oscillatory;
by 615F(a-vi), $\pmb u_{n+1}$ is moderately oscillatory.   Now if
$\sigma\in\Cal S_{n+1}$, we have

\Centerline{$\int_{(\Cal S_{n+1}\wedge\sigma)\wedge\tau_n}
   \bar h(\pmb u_{n+1},\pmb y)d\pmb v
=\int_{\Cal S_n\wedge(\sigma\wedge\tau_n)}
   \bar h(\pmb u_n,\pmb y\restr\Cal S_n)d(\pmb v\restr\Cal S_n)
=u_{n,\sigma\wedge\tau_n}-u_{\star,\sigma\wedge\tau_n}$,}

\Centerline{$\int_{(\Cal S_{n+1}\wedge\sigma)\vee\tau_n}
  \bar h(\pmb u_{n+1},\pmb y)d\pmb v
=\int_{\tilde{\Cal S}_n\wedge(\sigma\vee\tau_n)}
   \bar h(\tildepmb u_n,\tildepmb y)d\tildepmb v
=\tilde u_{\sigma\vee\tau_n}-u_{\star,\sigma\vee\tau_n}
   -u_{n\tau_n}+u_{\star\tau_n}$.}

\noindent By 613J(c-i),

$$\eqalign{\int_{\Cal S_{n+1}\wedge\sigma}
   \bar h(\pmb u_{n+1},\pmb y)d\pmb v
&=u_{n,\sigma\wedge\tau_n}-u_{\star,\sigma\wedge\tau_n}
   +\tilde u_{\sigma\vee\tau_n}-u_{\star,\sigma\vee\tau_n}
   -u_{n\tau_n}+u_{\star\tau_n}
\cr&=u_{n+1,\sigma\wedge\tau_n}
   +u_{n+1,\sigma\vee\tau_n}-u_{n+1,\tau_n}
   -u_{\star,\sigma\wedge\tau_n}-u_{\star,\sigma\vee\tau_n}
   +u_{\star\tau_n}
\cr&=u_{n+1,\sigma}-u_{\star\sigma}
\cr}$$

\noindent by 612Df.   As $\sigma$ is arbitrary,

\Centerline{$\pmb u_{n+1}
=\pmb u_{\star}\restr\Cal S_{n+1}
  +ii_{\pmb v}(\bar h(\pmb u_{n+1},\pmb y))$}

\noindent and the induction continues.\ \Qed

\medskip

{\bf (e)} At the end of the induction set
$\Cal S_{\infty}=\bigcup_{n\in\Bbb N}\Cal S_n$;
as $\pmb u_{n+1}$ extends
$\pmb u_n$ for every $n$, we have a process
$\pmb u_{\infty}=\bigcup_{n\in\Bbb N}\pmb u_n$ with domain
$\Cal S_{\infty}$.
If $\sigma$, $\sigma'\in\Cal S_{\infty}$,
there is an $n\in\Bbb N$ such that
$\Cal S_n$ contains both, so $\pmb u_{\infty}$ is fully adapted.
It is locally moderately oscillatory
because if $\sigma\in\Cal S_{\infty}$
there is an $n\in\Bbb N$ such that $\sigma\in\Cal S_n$ and
$\pmb u_{\infty}\restr\Cal S_{\infty}\wedge\sigma
=\pmb u_n\restr\Cal S_n\wedge\sigma$
is near-simple.   Because $\tau_n\in\Cal S_{\infty}$ and

\Centerline{$\inf_{n\in\Bbb N}\sup_{\sigma\in\Cal S_{\infty}}
  \Bvalue{\tau_n<\sigma}
\Bsubseteq\inf_{n\in\Bbb N}\Bvalue{\tau_n<\max\Cal S}=0$,}

\noindent $\pmb u_{\infty}$ is moderately oscillatory (654Cb).

Again because $\inf_{n\in\Bbb N}\Bvalue{\tau_n<\max\Cal S}=0$,
$\Cal S_{\infty}$
covers $\Cal S$, so $\Cal S_{\infty}$ and $\Cal S$ have the same covered
envelope, and $\pmb u_{\infty}$ has an extension to a process
$\pmb u=\sfamilysS{u}$ with domain $\Cal S$ (612Qa).
Because $\pmb u\restr\Cal S_{\infty}$ is locally moderately oscillatory,
$\pmb u$ is locally moderately oscillatory (615F(b-v)), therefore moderately
oscillatory.

Now take any $n\in\Bbb N$ and $\sigma\in\Cal S_n$.   Then

\Centerline{$u_{\sigma}
=u_{n\sigma}
=u_{\star\sigma}
  +\int_{\Cal S_n\wedge\sigma}\bar h(\pmb u_n,\pmb y)d\pmb v
=u_{\star\sigma}
  +\int_{\Cal S\wedge\sigma}\bar h(\pmb u,\pmb y)d\pmb v$.}

\noindent So $\pmb u$ agrees with
$\pmb u_{\star}+ii_{\pmb v}(\bar h(\pmb u,\pmb v))$ on
$\Cal S_{\infty}$;   as $\Cal S_{\infty}$ covers $\Cal S$,
$\pmb u=\pmb u_{\star}+ii_{\pmb v}(\bar h(\pmb u,\pmb v))$.

\medskip

{\bf (f)} Now suppose that $\pmb u'=\sfamilysS{u'}$
is another moderately oscillatory process such that
$\pmb u'=\pmb u_{\star}+ii_{\pmb v}(\bar h(\pmb u',\pmb y))$.   Then
$\pmb u'\restr\Cal S_n=\pmb u_n$ for every $n\in\Bbb N$.
\Prf\ Induce on $n$.   If $n=0$ then $\Cal S_n=\{\min\Cal S\}$ and both
$\pmb u'$ and $\pmb u_{\star}$ take the value $u_{\star,\min\Cal S}$
at $\min\Cal S$.
For the inductive step to $n+1$, the inductive hypothesis tells us that
$u'_{\tau_n}=u_{n\tau_n}$.   Now, for
$\sigma\in\tilde{\Cal S}_n=[\tau_n,\tau_{n+1}]$,

$$\eqalign{u'_{\sigma}
&=u_{\star\sigma}
   +\int_{\Cal S\wedge\sigma}\bar h(\pmb u',\pmb y)d\pmb v
\cr&=u_{\star\sigma}
   +\int_{(\Cal S\wedge\sigma)\wedge\tau_n}\bar h(\pmb u',\pmb y)d\pmb v
   +\int_{(\Cal S\wedge\sigma)\vee\tau_n}\bar h(\pmb u',\pmb y)d\pmb v
\cr&=u_{\star\sigma}+u_{n\tau_n}-u_{\star\tau_n}
   +\int_{\tilde{\Cal S}_n\wedge\sigma}\bar h(\pmb u',\pmb y)d\pmb v.
\cr}$$

\noindent As $\sigma$ is arbitrary,

\Centerline{$\pmb u'\restr\tilde{\Cal S}_n
=\pmb u_{\star}\restr\tilde{\Cal S}_n
 +(u_{n\tau_n}-u_{\star\tau_n})\tbf 1^{\tilde{\Cal S}_n}
 +ii_{\pmb v\restr\tilde{\Cal S}_n}
  (\bar h(\pmb u'\restr\tilde{\Cal S}_n,\pmb y\restr\tilde{\Cal S}_n))$.}

\noindent But 654H assured us that this equation had a unique solution, so
$\pmb u'\restr\tilde{\Cal S}_n$ must be identical with $\tildepmb u_n$
as found in (c), and therefore agrees with $\pmb u_{n+1}$ and with
$\pmb u$ on $\tilde{\Cal S}_n$.   As $\Cal S_n\cup\tilde{\Cal S}_n$ covers
$\Cal S_{n+1}$, $\pmb u'$ agrees with $\pmb u$ on $\Cal S_{n+1}$.   So the
induction continues.\ \Qed

Accordingly $\pmb u'$ agrees with $\pmb u$ on $\Cal S_{\infty}$
and therefore on
its full envelope $\Cal S$.   Thus $\pmb u$ is uniquely defined.

\medskip

{\bf (g)} So we have the result in the special case in which $\Cal S$
has a greatest member and $\pmb w$ is a martingale.
For the general case, let $\Cal S'$ be a covering ideal of $\Cal S$
such that $\pmb w\restr\Cal S'$ is a martingale.
Then (b)-(f) tell us that for
every $\tau\in\Cal S'$ there is a unique near-simple
process $\pmb u_{\tau}$ with domain $\Cal S\wedge\tau$ such that
$\pmb u_{\tau}
=\pmb u_{\star}\restr\Cal S\wedge\tau
  +ii_{\pmb v\restr\Cal S\wedge\tau}
    (\bar h(\pmb u_{\tau},\pmb y\restr\Cal S\wedge\tau))$.
If $\tau\le\tau'$ in $\Cal S'$ then
$\pmb u_{\tau'}\restr\Cal S\wedge\tau$ satisfies the same equation as
$\pmb u_{\tau}$, so must be equal to $\pmb u_{\tau}$.   As in (e), with
$\Cal S'$ in place of $\Cal S_{\infty}$,
there is therefore a fully adapted process $\pmb u=\sfamilysS{u}$
with domain $\Cal S$ extending every $\pmb u_{\tau}$, and such that
$\pmb u=\pmb u_{\star}+ii_{\pmb v}(\bar h(\pmb u,d\pmb y))$.
Because $\pmb u\restr\Cal S\wedge\tau=\pmb u_{\tau}$ is near-simple for
every $\tau\in\Cal S'$, $\pmb u$ is locally near-simple (631F(b-v)).
And as in (e), we see that if $\pmb u'$ has the same property, then
$\pmb u'\restr\Cal S_{\tau}=\pmb u_{\tau}$ for every $\tau\in\Cal S'$, so
that $\pmb u'=\pmb u$.

This completes the proof.
}%end of proof of 654I

\leader{654J}{Lemma} Suppose that $\family{t}{T}{\frak A_t}$ is
right-continuous, and that $\Cal S=[\min\Cal S,\max\Cal S]$ is an
interval in $\Cal T$.   Define $\pmb z=\sfamilysS{z}$ by setting
$z_{\sigma}=\chi\Bvalue{\sigma<\max\Cal S}$ for $\sigma\in\Cal S$.
Suppose that $\pmb x\in\cmmnt{\MSi=\mskip5mu}\MSi(\Cal S)$,
and that $\pmb v\in\Mns=\Mns(\Cal S)$ is an integrator.

(a) $\pmb z\times\Siiv(\pmb x)
=\pmb z\times\Sii_{\pmb z\times\pmb v}(\pmb x)$.

(b) $\Siiv(\pmb x)_<=\Siiv(\pmb z\times\pmb x)_<$.

\proof{{\bf (a)} For any $\sigma\in\Cal S$,

$$\eqalignno{\Bvalue{\Sint_{\Cal S\wedge\sigma}\pmb x\,d\pmb v
  \ne\Sint_{\Cal S\wedge\sigma}\pmb x\,d(\pmb z\times\pmb v)}
&\Bsubseteq\Bvalue{\pmb v\restr\Cal S\wedge\sigma
  \ne(\pmb z\times\pmb v)\restr\Cal S\wedge\sigma}
\cr\displaycause{646C}&
\Bsubseteq\Bvalue{\max\Cal S\le\sigma}
\Bsubseteq\Bvalue{\chi\Bvalue{\sigma<\max\Cal S}=0}.
\cr}$$

\noindent So
$\pmb z\times(\Siiv(\pmb x)-\Sii_{\pmb z\times\pmb v}(\pmb x))=0$
and we have the result.

\medskip

{\bf (b)} For $\sigma\in\Cal S$, set
$u_{\sigma}
=\Sint_{\Cal S\wedge\sigma}\pmb x\,d\pmb v$ and
$u'_{\sigma}
=\Sint_{\Cal S\wedge\sigma}\pmb z\times\pmb x\,d\pmb v$.   Then

$$\eqalignno{
\Bvalue{z_{\sigma}\times u_{\sigma}\ne z_{\sigma}\times u'_{\sigma}}
&=\Bvalue{z_{\sigma}=1}\Bcap\Bvalue{u_{\sigma}\ne u'_{\sigma}}
\cr&\Bsubseteq\Bvalue{\sigma<\max\Cal S}
   \Bcap\Bvalue{\pmb x\restr\Cal S\wedge\sigma
       \ne(\pmb z\times\pmb x)\restr\Cal S\wedge\sigma}
\cr\displaycause{647J}&
\Bsubseteq\Bvalue{\sigma<\max\Cal S}
   \Bcap\sup_{\sigma'\in\Cal S\wedge\sigma}\Bvalue{z_{\sigma'}=0}
\cr&=\Bvalue{\sigma<\max\Cal S}
   \Bcap\sup_{\sigma'\in\Cal S\wedge\sigma}\Bvalue{\sigma'=\max\Cal S}
=0.
\cr}$$

\noindent So
$z_{\sigma}\times u_{\sigma}=z_{\sigma}\times u'_{\sigma}$;  as $\sigma$ is
arbitrary,
$\pmb z\times\Siiv(\pmb x)=\pmb z\times\Siiv(\pmb z\times\pmb x)$.
Putting this together with 654Ee,

\Centerline{$\Siiv(\pmb x)_<
=(\pmb z\times\Siiv(\pmb x))_<
=(\pmb z\times\Siiv(\pmb z\times\pmb x))_<
=\Siiv(\pmb z\times\pmb x)_<$.}
}%end of proof of 654J

\leader{654K}{}\cmmnt{ Now for an S-integral version of 654G.

\medskip

\noindent}{\bf Lemma}
Suppose $\family{t}{T}{\frak A_t}$ is right-continuous and that
$\Cal S=[\min\Cal S,\max\Cal S]$ is an interval in $\Cal T$.
Let $\Mns^0$ be the space of near-simple processes
$\pmb u=\sfamilysS{u}$ with domain $\Cal S$ such that $u_{\min\Cal S}=0$.
For a near-simple integrator $\pmb v$, write $\pmb v^*$
for its quadratic variation.
Suppose that $\pmb w$, $\pmb w'\in\Mns^0$ are such that
$\pmb w$ is a martingale and $\pmb w'$ is of bounded variation;  set
$\pmb v=\pmb w+\pmb w'$.   If
$\pmb x\in\cmmnt{\MSi^0=\mskip5mu}\MSi^0(\Cal S)$,
$\pmb u\in\cmmnt{\Mmo=\mskip5mu}\Mmo(\Cal S)$ and $|\pmb x|\le\pmb u_<$, then

\Centerline{$\|\sup|\Siiv(\pmb x)|\|_2
\le 2(\sqrt{\|\pmb w^*\|_{\infty}}
  +\|\int_{\Cal S}|d\pmb w'|\|_{\infty})\|\sup|\pmb u|\|_2$.}

\proof{{\bf (a)} Write $\gamma$ for
$2(\sqrt{\|\pmb w^*\|_{\infty}}
  +\|\int_{\Cal S}|d\pmb w'|\|_{\infty})\|\sup|\pmb u|\|_2$.   If this is
infinite, we can stop, so suppose that $\gamma$ is finite.

The set $\{u:u\in L^2$, $\|u\|_2\le\gamma\}$ is closed in $L^0$ for the
topology of convergence in measure (613Bc).   Next,
$\pmb y\mapsto\sup|\pmb y|:\Mob\to L^0$ is continuous for the ucp topology
on $\Mob=\Mob(\Cal S)$ and the topology of convergence in measure on
$L^0$ (615C(b-ii)),
so $\{\pmb y:\|\sup|\pmb y|\|_2\le\gamma\}$ is closed in $\Mob$ for
the ucp topology.

\medskip

{\bf (b)} By 646O, $\Siiv$ is continuous on
$A=\{\pmb x:\pmb x\in\MSi^0$, $|\pmb x|\le\pmb u_<\}$ for the
S-integration topology on $A$ and the ucp topology on $\Mob$, so

\Centerline{$C
=\{\pmb x':\pmb x'\in A$, $\|\sup|\Siiv(\pmb x')|\|_2\le\gamma\}$}

\noindent is relatively closed in $A$ for the S-integration topology.
We know also that, writing $\Mmo$ for
$\Mmo(\Cal S)$ and $A_0$ for
$\{\pmb u':\pmb u'\in\Mmo$, $|\pmb u'|\le\pmb u\}$,
$\{\pmb u'_<:\pmb u'\in A_0\}$
is dense in $A$ (645La), $\Siiv(\pmb u'_<)=ii_{\pmb v}(\pmb u')$ for every
$\pmb u'\in A_0$ (646Kc), and that

\Centerline{$\|\sup|ii_{\pmb v}(\pmb u')|\|_2
\le2(\sqrt{\|\pmb w^*\|_{\infty}}
  +\|\int_{\Cal S}|d\pmb w'|\|_{\infty})\|\sup|\pmb u'|\|_2
\le\gamma$}

\noindent for every $\pmb u'\in A_0$ (654G).   So $\pmb u'_<\in C$ for every
$\pmb u'\in A_0$, and $C$ is dense in $A$.   But this means that $C$ is
actually equal to $A$;  consequently $\pmb x\in C$ and
$\|\sup|\Siiv(\pmb x)|\|_2\le\gamma$, as claimed.
}%end of proof of 654K

\leader{654L}{Lemma} Suppose that $\family{t}{T}{\frak A_t}$ is
right-continuous.   Let $\Cal S=[\min\Cal S,\max\Cal S]$ be an interval in
$\Cal T$.   Suppose that $h:\BbbR^2\to\Bbb R$ is
a locally bounded Borel measurable function
and that $K\ge 0$ is such that
$|h(\alpha,\beta)-h(\alpha',\beta)|\le K|\alpha-\alpha'|$ for all $\alpha$,
$\alpha'$, $\beta\in\Bbb R$;  let $\pmb w=\sfamilysS{w}$,
$\pmb w'=\sfamilysS{w'}$ be near-simple
processes with domain $\Cal S$ such that $\pmb w$ is a martingale,
$\pmb w'$ is of bounded variation and
$w_{\min\Cal S}=w'_{\min\Cal S}=0$.
Write $\pmb w^*$ for the quadratic variation of $\pmb w$,
$\pmb w^{\prime\uparrow}$ for the cumulative variation of
$\pmb w'$, and $\pmb z$ for
$\family{\sigma}{\Cal S}{\chi\Bvalue{\sigma<\max\Cal S}}$.   Suppose that
$2K(\sqrt{\|\pmb w^*\|_{\infty}}
  +2\|\pmb z\times\pmb w^{\prime\uparrow}\|_{\infty})<1$.   Set
$\pmb v=\pmb w+\pmb w'$.   Then for any $\pmb x_{\star}$,
$\pmb y\in\MSi=\MSi(\Cal S)$ there is a unique
process $\pmb x\in\MSi$ such that

\Centerline{$\pmb x
=\pmb x_{\star}+\Siiv(\bar h(\pmb x,\pmb y))_<$.}

\cmmnt{\medskip

\noindent{\bf Remark} I ought to say at once that by
$\Siiv(\bar h(\pmb x,\pmb y))_<$ I mean the previsible version
$(\Siiv(\bar h(\pmb x,\pmb y)))_<$ of the indefinite S-integral
$\Siiv(\bar h(\pmb x,\pmb y))$.   I think it looks better without the
extra brackets.}

\proof{ We can follow the same line of argument as in 654H.

\medskip

{\bf (a)} Suppose that $\|\pmb x_{\star}\|_{\infty}$ and
$\|\pmb y\|_{\infty}$ are
both finite, and that $2K\gamma<1$ where $\gamma=
\sqrt{\|\pmb w^*\|_{\infty}}+\|\int_{\Cal S}|d\pmb w'|\|_{\infty}$.

\medskip

\quad{\bf (i)} Define $\sequencen{\pmb x_n}$,
$\sequencen{\pmb u_n}$ by saying that $\pmb x_0=\pmb x_{\star}$ and

\Centerline{$\pmb u_n=\Siiv(\bar h(\pmb x_n,\pmb y))$,
\quad$\pmb x_{n+1}=\pmb x_{\star}+\pmb u_{n<}$}

\noindent for $n\in\Bbb N$.
Then $\bar h(\pmb x_n,\pmb y)\in\MSi$ (645Jb) and
$\pmb u_n$ belongs to $\Mns=\Mns(\Cal S)$ (646N), so
$\pmb x_{n+1}\in\MSi$, for any $n\ge 0$.

For $n\ge 0$,

$$\eqalignno{|\bar h(\pmb x_{n+1},\pmb y)
  -\bar h(\pmb x_n,\pmb y)|
&\le K|\pmb x_{n+1}-\pmb x_n|
=K|(\pmb u_n-\pmb u_{n-1})_<|
\Displaycause{counting $\pmb u_{-1}$ as $\tbf 0$}
=|K(\pmb u_n-\pmb u_{n-1})|_<\cr}$$

\noindent (641Gd), so

$$\eqalignno{\|\sup|\pmb u_{n+1}-\pmb u_n|\|_2
&=\|\sup|\Siiv(\bar h(\pmb x_{n+1},\pmb y)
  -\bar h(\pmb x_n,\pmb y))|\|_2
\cr&\le 2\gamma\|\sup|K(\pmb u_n-\pmb u_{n-1})|\|_2
\cr\displaycause{654K}&
=2\gamma K\|\sup|\pmb u_n-\pmb u_{n-1}|\|_2
\cr}$$

\noindent for $n\in\Bbb N$.   At the beginning of the iteration,
because $h$ is locally bounded and both $\pmb x_{\star}$ and
$\pmb y$ are
bounded, $K'=\|\bar h(\pmb x_{\star},\pmb y)\|_{\infty}$ is
finite, $|\bar h(\pmb x_{\star},\pmb y)\times\tbfoneminus{\Cal S}|
\le K'\tbfoneminus{\Cal S}$ and

$$\eqalign{\|\sup|\pmb u_0-\pmb u_{-1}|\|_2
&=\|\sup|\Siiv(\bar h(\pmb x_{\star},\pmb y))|\|_2
\cr&=\|\sup|\Siiv(\bar h(\pmb x_{\star},\pmb y)\times\tbfoneminus{\Cal S})|\|_2
\le 2\gamma\|K'\chi 1\|_2
=2K'\gamma\cr}$$

\noindent is finite.   As $2K\gamma<1$,
$\sum_{n=0}^{\infty}\|\sup|\pmb u_{n+1}-\pmb u_n|\|_2$ is finite.

\medskip

\quad{\bf (ii)} As in 654H, it follows that
$\sequencen{\pmb u_n}$ is Cauchy for the ucp uniformity,
and has a limit $\pmb u$ in $\Mns$.   This time, we need to check that
$\lim_{n\to\infty}\|\sup|\pmb u_n-\pmb u|\|_2=0$.   \Prf\ For any $n$,

\Centerline{$\sup|\pmb u-\pmb u_n|\le\sum_{i=n+1}^{\infty}\sup|u_i-u_{i-1}|$,}

\Centerline{$\|\sup|\pmb u-\pmb u_n|\|_2
\le\sum_{i=n+1}^{\infty}\|\sup|u_i-u_{i-1}|\|_2\to\infty$}

\noindent as $n\to\infty$.\ \Qed

Set
$\pmb x=\pmb x_{\star}+\pmb u_<$.   For $n\ge 0$,
$\pmb x-\pmb x_{n+1}=(\pmb u-\pmb u_n)_<$, so

\Centerline{$|\bar h(\pmb x,\pmb y)-\bar h(\pmb x_{n+1},\pmb y)|
\le K|\pmb u-\pmb u_n|_<$,}

$$\eqalign{\|\sup|\Siiv(\bar h(\pmb x,\pmb y)))
  -\pmb u_{n+1}|\|_2
&=\|\sup|\Siiv(\bar h(\pmb x,\pmb y)
  -\bar h(\pmb x_{n+1},\pmb y))|\|_2
\cr&\le 2K\gamma\|\sup|\pmb u-\pmb u_n|\|_2
\to 0
\cr}$$

\noindent as $n\to\infty$.   Accordingly, taking limits for the ucp
topology,

\Centerline{$\pmb x-\pmb x_{\star}=\pmb u_<
=(\lim_{n\to\infty}\pmb u_n)_<=\Siiv(\bar h(\pmb x,\pmb y))_<$}

\noindent and $\pmb x=\pmb x_{\star}+\Siiv(\bar h(\pmb x,\pmb y))_<$.

\medskip

{\bf (b)} Once again assume that $2K\gamma<1$ where
$\gamma=
\sqrt{\|\pmb w^*\|_{\infty}}+\|\int_{\Cal S}|d\pmb w'|\|_{\infty}$.
Suppose that

\Centerline{$\pmb x_{0\star}$, $\pmb x_{1\star}$,
$\pmb y_0$, $\pmb y_1$, $\pmb x_0$, $\pmb x_1\in\MSi$}

\noindent are such that

\Centerline{$\pmb x_0=\pmb x_{0\star}+\Siiv(\bar h(\pmb x_0,\pmb y_0))_<$,
\quad$\pmb x_1=\pmb x_{1\star}+\Siiv(\bar h(\pmb x_1,\pmb y_1))_<$.}

\noindent Set

\Centerline{$\pmb u_0=\Siiv(\bar h(\pmb x_0,\pmb y_0))$,
\quad$\pmb u_1=\Siiv(\bar h(\pmb x_1,\pmb y_1))$,}

\noindent and suppose that $\sup|\pmb u_0|\in L^2$.   Set
$a=\Bvalue{\pmb x_{0\star}\ne\pmb x_{1\star}}
\Bcup\Bvalue{\pmb y_0\ne\pmb y_1}$.
Then $\Bvalue{\pmb u_0\ne\pmb u_1}\Bsubseteq a$.
\Prf\ Let $\epsilon>0$.
Let $M\ge 0$ be such that $\bar\mu\Bvalue{\sup|\pmb u_1|>M}\le\epsilon$
and set $a'=a\Bcup\Bvalue{\sup|\pmb u_1|>M}$.

Now repeat the inductive construction of (a-i), adjusted as follows.
Set $\tildepmb u_0
=\med(-M\tbf 1^{(\Cal S)},\penalty-100\pmb u_1,M\tbf 1^{(\Cal S)})$ and

\Centerline{$\tildepmb x_{n+1}
=\pmb x_{0\star}+\tildepmb u_{n-}$,
\quad$\tildepmb u_{n+1}=\Siiv(\bar h(\tildepmb x_{n+1},\pmb y_0))$}

\noindent for $n\in\Bbb N$.   Then

\Centerline{$|\bar h(\tildepmb x_{n+1},\pmb y_0)
  -\bar h(\pmb x_0,\pmb y_0)|
\le K|\tildepmb x_{n+1}-\pmb x_0|
=K|\tildepmb u_n-\pmb u_0|_<$,}

$$\eqalign{\|\sup|\tildepmb u_{n+1}-\pmb u_0|\|_2
&=\|\sup|\Siiv(\bar h(\tildepmb x_{n+1},\pmb y_0)
  -\bar h(\pmb x_0,\pmb y_0))|\|_2
\cr&\le 2K\gamma\|\sup|\tildepmb u_n-\pmb u_0|\|_2
\cr}$$

\noindent for each $n$, while
$\|\sup|\tildepmb u_0-\pmb u_0|\|_2\le M+\|\sup|\pmb u_0|\|_2$
is finite.   So
$\lim_{n\to\infty}\|\sup|\tildepmb u_n-\pmb u_0|\|_2=0$ and
$\pmb u_0=\lim_{n\to\infty}\tildepmb u_n$ for the ucp topology.

At the same time, we find that
$\Bvalue{\tildepmb u_n\ne\pmb u_1}\Bsubseteq a'$ for every $n$.
To see this, induce on $n$.   At the start,
$\Bvalue{\tildepmb u_0\ne\pmb u_1}
=\Bvalue{\sup|\pmb u_1|>M}\Bsubseteq a'$.   For the inductive step,

$$\eqalignno{\Bvalue{\tildepmb u_{n+1}\ne\pmb u_1}
&=\Bvalue{\Siiv(\bar h(\tildepmb x_{n+1},\pmb y_0)
  -\bar h(\pmb x_1,\pmb y_1))\ne\tbf 0}
\Bsubseteq\Bvalue{\bar h(\tildepmb x_{n+1},\pmb y_0)
  -\bar h(\pmb x_1,\pmb y_1)\ne\tbf 0}
\cr\displaycause{647J}&
\Bsubseteq\Bvalue{\tildepmb x_{n+1}\ne\pmb x_1}
   \Bcup\Bvalue{\pmb y_0\ne\pmb y_1}\cr&
\Bsubseteq\Bvalue{\pmb x_{0\star}\ne\pmb x_{1\star}}
   \Bcup\Bvalue{\tildepmb u_{n<}\ne\pmb u_{1<}}
   \Bcup a
\Bsubseteq\Bvalue{\tildepmb u_n\ne\pmb u_1}\Bcup a
\Displaycause{641G(a-iii)}
\Bsubseteq a'
\cr}$$

\noindent by the inductive hypothesis, so the induction proceeds.

It follows that
$\Bvalue{\pmb u_0\ne\pmb u_1}\Bsetminus a\Bsubseteq a'\Bsetminus a$ has
measure at most $\epsilon$.   As $\epsilon$ is arbitrary,
$\Bvalue{\pmb u_0\ne\pmb u_1}\Bsubseteq a$.\ \Qed

\medskip

{\bf (c)} For the second stage of the existence proof,
continue to assume that $2K\gamma<1$ where $\gamma=
\sqrt{\|\pmb w^*\|_{\infty}}+\|\int_{\Cal S}|d\pmb w'|\|_{\infty}$,
but drop the $\|\,\|_{\infty}$-boundedness conditions on
$\pmb x_{\star}$ and
$\pmb y$, and suppose only that $\pmb x_{\star}$, $\pmb y\in\MSi$.

\medskip

\quad{\bf (i)} In this case, set

\Centerline{$\pmb x_{k*}
=\med(-k\tbfoneminus{\Cal S},\pmb x_{\star},k\tbfoneminus{\Cal S})$,
\quad$\pmb y_k
=\med(-k\tbfoneminus{\Cal S},\pmb y,k\tbfoneminus{\Cal S})$,}

\Centerline{$a_k
=\Bvalue{\sup|\pmb x_{\star}|>k}\Bcup\Bvalue{\sup|\pmb y|>k}$}

\noindent for $k\in\Bbb N$.   (Recall that we know from
645Kb that $\pmb x_{\star}$ and $\pmb y$ are order-bounded.)
By (a), there is for each
$k$ a $\pmb u_k\in\Mns$ such that $\sup|\pmb u_k|$ is square-integrable
and, setting $\pmb x_k=\pmb x_{k*}+(\pmb u_k)_<$,
$\pmb u_k=\Siiv(\bar h(\pmb x_k,\pmb y_k))$.
By (b),   $\Bvalue{\pmb u_k\ne\pmb u_l}\Bsubseteq a_k$ for
$l\ge k$.   Since $\sequence{k}{a_k}$ is a non-increasing sequence with
infimum $0$,
$z=\sup_{k\in\Bbb N}\sup|\pmb u_k|
=\sup_{k\in\Bbb N}\chi(a_k\Bsetminus\chi a_{k+1})\times\sup|\pmb u_k|$
is defined in $L^0$ and
$\sup|\pmb u-\pmb u_k|\le z\times\chi a_k$ for each $k$.   This shows
that $\pmb u=\lim_{k\to\infty}\pmb u_k$ for the ucp topology, and
$\pmb u$ is near-simple (631B).
Next, setting $\pmb x=\pmb x_{\star}+\pmb u_<$, we have, for any $k$,

\Centerline{$\Bvalue{\pmb x\ne\pmb x_k}
\Bsubseteq\Bvalue{\pmb x_{\star}\ne\pmb x_k}
   \Bcup\Bvalue{\pmb u_<\ne\pmb u_{k-}}
\Bsubseteq a_k\Bcup\Bvalue{\pmb u\ne\pmb u_k}
=a_k$,}

$$\eqalign{\Bvalue{(\pmb u_k)_<\ne\Siiv(\bar h(\pmb x,\pmb y))_<}
&\Bsubseteq\Bvalue{\pmb u_k\ne\Siiv(\bar h(\pmb x,\pmb y))}
=\Bvalue{\Siiv(\bar h(\pmb x_k,\pmb y_k))
  \ne\Siiv(\bar h(\pmb x,\pmb y))}
\cr&\Bsubseteq\Bvalue{\bar h(\pmb x_k,\pmb y_k)\ne\bar h(\pmb x,\pmb y)}
\Bsubseteq\Bvalue{\pmb x_k\ne\pmb x}\Bcup\Bvalue{\pmb y_k\ne\pmb y}
\Bsubseteq a_k
\cr}$$

\noindent and

\Centerline{$\Bvalue{\pmb x\ne\pmb x_{\star}
  +\Siiv(\bar h(\pmb x,\pmb y))_<}
\Bsubseteq\Bvalue{\pmb x\ne\pmb x_k}
  \Bcup\Bvalue{\pmb x_{\star}\ne\pmb x_{k*}}
  \Bcup\Bvalue{\Siiv(\bar h(\pmb x,\pmb y))_<\ne(\pmb u_k)_<}
\Bsubseteq a_k$.}

\noindent As $k$ is arbitrary,
$\pmb x=\pmb x_{\star}+\Siiv(\bar h(\pmb x,\pmb y))_<$.

\medskip

\quad{\bf (ii)} To see that $\pmb x$ is unique, let $\pmb x'\in\MSi$ be
such that $\pmb x'=\pmb x_{\star}+\Siiv(\bar h(\pmb x',\pmb y))_<$.
This time, (b) tells us that, setting
$\pmb u'=\Siiv(\bar h(\pmb x',\pmb y))$,
$\Bvalue{\pmb u'\ne\pmb u_k}\Bsubseteq a_k$ for each $k$.
So $\pmb u'=\pmb u$ and $\pmb x'=\pmb x$.

\medskip

{\bf (d)} Now turn to the given hypothesis that
$2K(\sqrt{\|\pmb w^*\|_{\infty}}
  +2\|\pmb z\times\pmb w^{\prime\uparrow}\|_{\infty})<1$.

\medskip

\quad{\bf (i)} As in 654H, set $\pmb w_1'=\pmb z\times\pmb w'$ and
$\pmb v_1=\pmb w+\pmb w_1'$.   Then, as before, we have
$2K(\sqrt{\|\pmb w^*\|_{\infty}}+\|\int_{\Cal S}|d\pmb w_1'|\|_{\infty})
<1$.   So (c-i) tells us that
there is an S-integrable process $\pmb x_1$ such that
$\pmb x_1=\pmb x_{\star}+\Sii_{\pmb v_1}(\bar h(\pmb x_1,\pmb y))_<$.
Set $\pmb w_2'=\pmb w'-\pmb w_1'$ and $\pmb x
=\pmb x_1+\Sii_{\pmb w_2'}(\bar h(\pmb x_1,\pmb y))_<$.
We see that

\Centerline{$\pmb v_1+\pmb w_2'
=\pmb w+\pmb w'=\pmb v$,}

\Centerline{$\pmb x
=\pmb x_{\star}+\Sii_{\pmb v_1}(\bar h(\pmb x_1,\pmb y))_<
     +\Sii_{\pmb w_2'}(\bar h(\pmb x_1,\pmb y))_<
=\pmb x_{\star}+\Sii_{\pmb v}(\bar h(\pmb x_1,\pmb y))_<$,}

$$\eqalignno{\pmb z\times\pmb x
&=\pmb z\times\pmb x_1
   +\pmb z\times\Sii_{\pmb w_2'}(\bar h(\pmb x_1,\pmb y))_<
=\pmb z\times\pmb x_1
   +\pmb z\times\Sii_{\pmb z\times\pmb w_2'}
     (\bar h(\pmb x_1,\pmb y))_<
\cr\displaycause{654Ja}&
=\pmb z\times\pmb x_1
\cr}$$

\noindent because $\pmb z\times\pmb w_2'=0$.   Consequently

$$\eqalignno{\Siiv(\bar h(\pmb x,\pmb y))_<
&=\Siiv(\pmb z\times\bar h(\pmb x,\pmb y))_<
\cr\displaycause{654Jb)}&
=\Siiv(\pmb z\times\bar h(\pmb z\times\pmb x,\pmb y))_<
\cr\displaycause{619Ge again, as in the proof of 654H}&
=\Siiv(\pmb z\times\bar h(\pmb z\times\pmb x_1,\pmb y))_<
=\Siiv(\bar h(\pmb x_1,\pmb y))_<
\cr}$$

\noindent and
$\pmb x=\pmb x_{\star}+\Siiv(\bar h(\pmb x,\pmb y))_<$, as required.

\medskip

\quad{\bf (ii)}
To show uniqueness we can run the argument backwards, as follows.
Suppose that $\pmb x'\in\MSi$ is such that
$\pmb x'
=\pmb x_{\star}+\Sii_{\pmb v}(\bar h(\pmb x',\pmb y))_<$.
Set $\pmb x_1'
=\pmb x_{\star}+\Sii_{\pmb v_1}(\bar h(\pmb x',\pmb y))_<$.
Then

$$\eqalign{\pmb z\times\pmb x_1'
&=\pmb z\times\pmb x_{\star}
  +\pmb z\times\Sii_{\pmb z\times\pmb v_1}
    (\bar h(\pmb x',\pmb y))_<
\cr&=\pmb z\times\pmb x_{\star}
  +\pmb z\times\Sii_{\pmb z\times\pmb v}
    (\bar h(\pmb x',\pmb y))_<
=\pmb z\times\pmb x'
\cr}$$

\noindent so that
$\Sii_{\pmb v_1}(\bar h(\pmb x_1',\pmb y))_<
=\Sii_{\pmb v_1}(\bar h(\pmb x',\pmb y))_<$ and
$\pmb x_1'
=\pmb x_{\star}+\Sii_{\pmb v_1}(\bar h(\pmb x_1',\pmb y))_<$.
But this implies that $\pmb x_1'=\pmb x_1$, by the uniqueness
established in (c-ii), and $\pmb z\times\pmb x_1'=\pmb z\times\pmb x_1$.
Now

$$\eqalign{\pmb x'
&=\pmb x_1'+\Sii_{\pmb w_2'}(\bar h(\pmb x',\pmb y))_<
=\pmb x_1'+\Sii_{\pmb w_2'}(\bar h(\pmb x_1',\pmb y))_<
\cr&=\pmb x_1+\Sii_{\pmb w_2'}(\bar h(\pmb x_1,\pmb y))_<
=\pmb x.\cr}$$

\noindent Thus the solution offered in (i) is unique.
}%end of proof of 654L

\leader{654M}{}\cmmnt{ The key step in the next stage involves
re-starting an S-integral, so I separate it out.

\medskip

\noindent}{\bf Lemma} Suppose that $\family{t}{T}{\frak A_t}$
is right-continuous, and that $\Cal S\subseteq\Cal T$ is an order-convex
sublattice
with a least member.   Let $h:\BbbR^2\to\Bbb R$ be a locally bounded
Borel measurable function.
Take processes $\pmb x$, $\pmb x_{\star}$, $\pmb y\in\MSi(\Cal S)$ and
an integrator $\pmb v\in\Mns(\Cal S)$.   Set
$\pmb u=\Siiv(\bar h(\pmb x,\pmb y))$ and express $\pmb u$ as
$\sfamilysS{u}$, $\pmb u_<$ as $\family{\sigma}{\Cal S}{u_{<\sigma}}$.

Fix $\tau\in\Cal S$.   Set

\Centerline{$\Cal S'=\Cal S\wedge\tau$,
\quad$\pmb x'=\pmb x\restr\Cal S'$,
\quad$\pmb v'=\pmb v\restr\Cal S'$,
\quad$\pmb y'=\pmb y\restr\Cal S'$,
\quad$\pmb x'_{\star}=\pmb x_{\star}\restr\Cal S'$,}

\Centerline{$\Cal S''=\Cal S\vee\tau$,
\quad$\pmb x''=\pmb x\restr\Cal S''$,
\quad$\pmb v''=\pmb v\restr\Cal S''$,
\quad$\pmb y''=\pmb y\restr\Cal S''$
\quad$\pmb x''_{\star}=\pmb x_{\star}\restr\Cal S''+\tildepmb x_*$}

\noindent where $\tildepmb x_*
=u_{<\tau}\tbf 1^{(\Cal S'')}+(u_{\tau}-u_{<\tau})\tbf 1^{(\Cal S'')}_<$.

(a) $\Sii_{\pmb v'}(\bar h(\pmb x',\pmb y'))
=\Siiv(\bar h(\pmb x,\pmb y))\restr\Cal S'$, so
$\Sii_{\pmb v'}(\bar h(\pmb x',\pmb y'))_<
=\Siiv(\bar h(\pmb x,\pmb y))_<\restr\Cal S'$.

(b) $\pmb x''_{\star}\in\MSi(\Cal S'')$.

(c) $\pmb x=\pmb x_{\star}+\Siiv(\bar h(\pmb x,\pmb y))_<$ if and only if

\Centerline{$\pmb x'
=\pmb x'_{\star}+\Sii_{\pmb v'}(\bar h(\pmb x',\pmb y'))_<$\quad and\quad
$\pmb x''=\pmb x''_{\star}
  +\Sii_{\pmb v''}(\bar h(\pmb x'',\pmb y''))_<$.}

\proof{{\bf (a)(i)} We should start by noting that
$\bar h(\pmb x,\pmb y)$ is S-integrable, by 645J as usual,
so we can talk about the indefinite S-integral
$\Siiv(\bar h(\pmb x,\pmb y))$.   Similarly, because
$\pmb x'$, $\pmb y'$, $\pmb x''$ and $\pmb y''$
are S-integrable (646Hd) and $\pmb v'$, $\pmb v''$ are near-simple
integrators (631Fa, 616P(b-ii)), we have the indefinite S-integrals
$\Sii_{\pmb v'}(\bar h(\pmb x',\pmb y'))$  and
$\Sii_{\pmb v''}(\bar h(\pmb x'',\pmb y''))$.

For $\sigma\in\Cal S$ write $\Cal S_{\sigma}$ for $\Cal S\wedge\sigma$.

\medskip

\quad{\bf (ii)} If $\sigma\in\Cal S'$ then

$$\eqalignno{\Sint_{\Cal S_{\sigma}}\bar h(\pmb x,\pmb y)\,d\pmb v
&=\Sint_{\Cal S_{\sigma}}
   \bar h(\pmb x\restr\Cal S_{\sigma},\pmb y\restr\Cal S_{\sigma})
   \,d(\pmb v\restr\Cal S_{\sigma})
\cr&=\Sint_{\Cal S_{\sigma}}
   \bar h(\pmb x'\restr\Cal S_{\sigma},\pmb y'\restr\Cal S_{\sigma})
   \,d(\pmb v'\restr\Cal S_{\sigma})
\cr&=\Sint_{\Cal S_{\sigma}}
   \bar h(\pmb x',\pmb y')\,d\pmb v'.
\cr}$$

\noindent So $\Sii_{\pmb v'}(\bar h(\pmb x',\pmb y'))
=\Siiv(\bar h(\pmb x,\pmb y))\restr\Cal S'$.
By 641G(c-ii), it follows that $\Sii_{\pmb v'}(\bar h(\pmb x',\pmb y'))_<
=\Siiv(\bar h(\pmb x,\pmb y))_<\restr\Cal S'$.

\medskip

{\bf (b)} Here I should point out that
$\tildepmb x_*\times\tbfoneminus{\Cal S''}
=(u_{\tau}\tbf 1^{(\Cal S'')})_<$ belongs to $\MSi^0(\Cal S'')$ so
$\pmb x''_{\star}\in\MSi(\Cal S'')$.

\medskip

{\bf (c)(i)} For $\sigma\in\Cal S''$, set
$u''_{\sigma}
=\Sint_{[\tau,\sigma]}\bar h(\pmb x,\pmb y)\,d\pmb v$, so that
$\Sii_{\pmb v''}(\bar h(\pmb x'',\pmb y''))
=\family{\sigma}{\Cal S''}{u''_{\sigma}}$ and
$\Sii_{\pmb v''}(\bar h(\pmb x'',\pmb y''))_<
=\family{\sigma}{\Cal S''}{u''_{<\sigma}}$.

If $\sigma\in\Cal S''$, then

$$\eqalignno{u_{\sigma}
&=\Sint_{\Cal S_{\sigma}}\bar h(\pmb x,\pmb y)\,d\pmb v
=\Sint_{\Cal S'}\bar h(\pmb x,\pmb y)\,d\pmb v
 +\Sint_{[\tau,\sigma]}\bar h(\pmb x,\pmb y)\,d\pmb v
\cr\displaycause{646J, because $\Cal S_{\sigma}\wedge\tau=\Cal S'$ and
$\Cal S_{\sigma}\vee\tau=[\tau,\sigma]$}&
=u_{\tau}
 +\Sint_{[\tau,\sigma]}\bar h(\pmb x,\pmb y)\,d\pmb v
=u_{\tau}+u''_{\sigma}.
\cr}$$

\noindent We know that
$(\pmb u\restr\Cal S'')_<
=(\pmb u_<\restr\Cal S'')\times\tbfoneminus{\Cal S''}$ (641G(c-ii)),
so that

\Centerline{$\Bvalue{\tau<\sigma}
\Bsubseteq\Bvalue{u_{<\sigma}=u_{\tau}+u''_{<\sigma}}
  \Bcap\Bvalue{\tilde x_{*\sigma}=u_{\tau}}
\Bsubseteq\Bvalue{u_{<\sigma}=\tilde x_{*\sigma}+u''_{<\sigma}}$,}

\noindent where
$\tildepmb x_*=\langle\tilde x_{*\sigma}\rangle_{\sigma\in\Cal S''}$, while

\Centerline{$\Bvalue{\tau=\sigma}
\Bsubseteq\Bvalue{u_{<\sigma}=u_{<\tau}}
   \Bcap\Bvalue{u''_{<\sigma}=0}
   \Bcap\Bvalue{\tilde x_{*\sigma}=u_{<\tau}}
\Bsubseteq\Bvalue{u_{<\sigma}=\tilde x_{*\sigma}+u''_{<\sigma}}$.}

\noindent Thus $u_{<\sigma}=\tilde x_{*\sigma}+u''_{<\sigma}$;  as
$\sigma$ is arbitrary, $\Siiv(\bar h(\pmb x,\pmb y))_<\restr\Cal S''
=\tildepmb x_*+\Sii_{\pmb v''}(\bar h(\pmb x'',\pmb y''))_<$.

\medskip

\quad{\bf (ii)} Writing $\pmb u'$ for
$\Sii_{\pmb v'}(\bar h(\pmb x',\pmb y'))$ we know from (a) that
$\pmb u'_<=\pmb u_<\restr\Cal S'$ and from (i) just above that
$\tildepmb x_*+\pmb u''_<=\pmb u_<\restr\Cal S''$.   Since
$\Cal S'\cup\Cal S''$ covers $\Cal S$, and $\pmb x$ and
$\pmb x_{\star}+\pmb u_<$ are fully adapted,

$$\eqalign{&\pmb x=\pmb x_{\star}+\pmb u_<
\cr&\mskip40mu
\iff\pmb x\restr\Cal S'=\pmb x_{\star}\restr\Cal S'+\pmb u_<\restr\Cal S'
   \text{ and }
    \pmb x\restr\Cal S''=\pmb x_{\star}\restr\Cal S''+\pmb u_<\restr\Cal S''
\cr&\mskip40mu
\iff\pmb x'=\pmb x'_{\star}+\pmb u'_<
   \text{ and }
    \pmb x''=\pmb x_{\star}\restr\Cal S''+\tildepmb x_*+\pmb u''_<
\cr&\mskip40mu
\iff\pmb x'=\pmb x'_{\star}+\pmb u'_<
   \text{ and }
    \pmb x''=\pmb x''_{\star}+\pmb u''_<,
\cr}$$

\noindent that is,

$$\eqalign{
&\pmb x=\pmb x_{\star}+\Siiv(\bar h(\pmb x,\pmb y))_<
\cr&\mskip40mu
\iff\pmb x'=\pmb x'_{\star}+\Sii_{\pmb v'}(\bar h(\pmb x',\pmb y'))_<
\text{ and }
\pmb x''=\pmb x''_{\star}+\tildepmb x_*
  +\Sii_{\pmb v''}(\bar h(\pmb x'',\pmb y''))_<.
\cr}$$
}%end of proof of 654M

\leader{654N}{}\cmmnt{ With this in hand, we can embark on a full-strength
version of Theorem 654I.

\medskip

\noindent}{\bf Theorem} Suppose that $\family{t}{T}{\frak A_t}$ is
right-continuous.   Let $\Cal S$ be an order-convex sublattice of
$\Cal T$ with a least member.   Suppose that $h:\BbbR^2\to\Bbb R$ is
a locally bounded Borel measurable function
and that $K\ge 0$ is such that
$|h(\alpha,\beta)-h(\alpha',\beta)|\le K|\alpha-\alpha'|$ for all $\alpha$,
$\alpha'$, $\beta\in\Bbb R$.   Let $\pmb v$ be a locally near-simple
local integrator
with domain $\Cal S$.   Then for any locally S-integrable processes
$\pmb x_{\star}\cmmnt{\mskip5mu=\family{\sigma}{\Cal S}{x_{\star\sigma}}}$,
$\pmb y$ with domain $\Cal S$ there is a unique
locally S-integrable process $\pmb x$ with domain $\Cal S$ such that

\Centerline{$\pmb x
=\pmb x_{\star}+\Siiv(\bar h(\pmb x,\pmb y))_<$.}

\proof{{\bf (a)} As in part (a) of the proof of 654I,
we can express $\pmb v$ as $\pmb w+\pmb w'$ where
$\pmb w=\sfamilysS{w}$ is a local martingale,
$\pmb w'=\sfamilysS{w'}$ is of bounded variation and
$\Osclln(\pmb w)\le\epsilon\chi 1$.
For the time being (down to the end of (c) below),
suppose that $\Cal S$ has a greatest member, so that
$\pmb v$ is near-simple and $\pmb x_{\star}$
and $\pmb y$ are S-integrable, and that $\pmb w$ is a martingale.

Let $\epsilon>0$ be such that $2K(\sqrt{\epsilon}+3\epsilon)<1$.
We know that $\pmb w$ and $\pmb w'$ are near-simple, so the cumulative
variation $\pmb w^{\prime\uparrow}=\sfamilysS{w^{\prime\uparrow}}$ of
$\pmb w'$ and the quadratic variation $\pmb w^*=\sfamilysS{w^*}$ of
$\pmb w$ are near-simple.

As in part (c) of the proof of 654I, there is a non-decreasing
sequence $\sequencen{\tau_n}$ in $\Cal S$ such that $\tau_0=\min\Cal S$,
$\inf_{n\in\Bbb N}\Bvalue{\tau_n<\max\Cal S}=0$ and whenever $n\in\Bbb N$
and $\sigma\in[\tau_n,\tau_{n+1}]$,

\Centerline{$\Bvalue{w^{\prime\uparrow}_{\sigma}+w^*_{\sigma}
   -w^{\prime\uparrow}_{\tau_n}-w^*_{\tau_n}
  \ge\epsilon}
\Bsubseteq\Bvalue{\sigma=\tau_{n+1}}$.}

\medskip

{\bf (b)} For $n\in\Bbb N$, write
$\Cal S_n=\Cal S\wedge\tau_n=[\min\Cal S,\tau_n]$.   Then we
have a sequence $\sequencen{\pmb x_n}$ of S-integrable processes such that

\Centerline{$\pmb x_n\in\MSi(\Cal S_n)$,
\quad$\pmb x_n=\pmb x_{\star}\restr\Cal S_n
  +\Sii_{\pmb v\restr\Cal S_n}
    (\bar h(\pmb x_n,\pmb y\restr\Cal S_n))_<$}

\noindent for each $n$.   \Prf\ Induce on $n$.   As in part (d) of the
proof of 654I, the induction starts with
$\pmb x_0=\pmb x_{\star}\restr\{\min\Cal S\}$.
For the inductive step to $n+1$, given
$\pmb x_n$, set $\pmb u
=\Sii_{\pmb v\restr\Cal S_n}(\bar h(\pmb x_n,\pmb y\restr\Cal S_n))$
and express $\pmb u$ as
$\family{\sigma}{\Cal S_n}{u_{\sigma}}$;  set

\Centerline{$\tilde{\Cal S}_n=[\tau_n,\tau_{n+1}]$,
\quad$\tildepmb x_0=\pmb x_{\star}\restr\tilde{\Cal S}_n
  +u_{<\tau_n}\tbf 1^{(\tilde{\Cal S}_n)}
  +(u_{\tau_n}-u_{<\tau_n})\tbfoneminus{\tilde{\Cal S}_n}$,}

\Centerline{$\tildepmb w
  =\pmb w\restr\tilde{\Cal S}_n
    -w'_{\tau_n}\tbf 1^{(\tilde{\Cal S}_n)}$,
\quad$\tildepmb w'=\pmb w'\restr\tilde{\Cal S}_n
  -w_{\tau_n}\tbf 1^{(\tilde{\Cal S}_n)}$,
\quad$\tildepmb v=\tildepmb w+\tildepmb w'$.}

\noindent Then $\tildepmb w$ and $\tildepmb w'$ are near-simple
processes with domain $\tilde{\Cal S}_n$, $\tildepmb w$ is a
martingale, $\tildepmb w'$ is of bounded variation
and both start with value $0$ at $\min\tilde{\Cal S}_n=\tau_n$.
The quadratic variation $\tildepmb w{}^*$
of $\tildepmb w$ is $\pmb w^*\restr\tilde{\Cal S}_n
  -w^*_{\tau_n}\tbf 1^{(\tilde{\Cal S}_n)}$,
and the cumulative variation $\tildepmb w^{\prime\uparrow}$
of $\tildepmb w'$ is $\pmb w^{\prime\uparrow}\restr\tilde{\Cal S}_n
  -w^{\uparrow}_{\tau_n}\tbf 1^{(\tilde{\Cal S}_n)}$.

As before, setting
$\pmb z=\family{\sigma}{\tilde{\Cal S}_n}{\chi\Bvalue{\sigma<\tau_{n+1}}}$,

\Centerline{$2K\bigl(\sqrt{\|\tildepmb w{}^*\|_{\infty}}
   +2\|\pmb z\times\tildepmb w^{\prime\uparrow}\|_{\infty}\bigr)
\le 2K(\sqrt{\epsilon}+3\epsilon)
<1$.}

\noindent We can therefore apply 654L to see
that there is a unique S-integrable process $\tildepmb x_n$ with domain
$\tilde{\Cal S}_n$ such that
$\tildepmb x_n=\tildepmb x_0
  +\Sii_{\tildepmb v}(\bar h(\tildepmb x_n,\tildepmb y))_<$.

Since the processes
$\pmb x_n$ and $\tildepmb x_n$ take the same value
$x_{\star\tau_n}+u_{<\tau_n}$ at $\tau_n$,
we have a fully adapted process $\pmb x_{n+1}$ with domain
$\Cal S_{n+1}$ extending them both.
By 646J, $\pmb x_{n+1}$ is S-integrable.   Now 654Ma tells us that
$\Sii_{\pmb v\restr\Cal S_n}
  (\bar h(\pmb x_{n+1}\restr\Cal S_n,\pmb y\restr\Cal S_n))=\pmb u$, and
654Mc tells us that

\Centerline{$\pmb x_{n+1}
=\pmb x_{\star}\restr\Cal S_{n+1}
  +\Sii_{\pmb v\restr\Cal S_{n+1}}
     (\bar h(\pmb x_{n+1},\pmb y\restr\Cal S_{n+1}))_<$,}

\noindent so the induction continues.\ \Qed

\medskip

{\bf (c)} At the end of the induction, as in part (e) of the proof of 654I,
we have a process $\pmb x=\sfamilysS{x}$ extending
every $\pmb x_n$.   Next, setting

\Centerline{$\pmb u_n
=\Siiv(\bar h(\pmb x\restr\Cal S_n,\pmb y\restr\Cal S_n))
=\Sii_{\pmb v\restr\Cal S_n}(\bar h(\pmb x_n,\pmb y\restr\Cal S_n))$,}

\noindent for each $n$, we see that $\pmb u_n$ is near-simple (646N)
and $\pmb u_n=\pmb u_{n+1}\restr\Cal S_n$ for each $n$.   Accordingly, just as
in the proof of 654I, there is a near-simple $\pmb u=\sfamilysS{u}$ such that
$\pmb u_n=\pmb u\restr\Cal S_n$ for every $n$.

Now $\pmb u_<$ is S-integrable (645R(a-i)) so $\pmb x_{\star}+\pmb u_<$
is S-integrable.   But

$$\eqalign{\pmb x\restr\Cal S_n&
=\pmb x_{\star}\restr\Cal S_n+\pmb u_{n<}
=\pmb x_{\star}\restr\Cal S_n+(\pmb u\restr\Cal S_n)_<\cr&
=\pmb x_{\star}\restr\Cal S_n+\pmb u_<\restr\Cal S_n
=(\pmb x_{\star}\restr\Cal S+\pmb u_<)\restr\Cal S_n
\cr}$$

\noindent for every $n$, by 641G(c-ii).   Since
$\bigcup_{n\in\Bbb N}\Cal S_n$ covers $\Cal S$,
$\pmb x=\pmb x_{\star}+\pmb u_<$ is S-integrable.   Consequently
$\bar h(\pmb x,\pmb y)$ is S-integrable (645J again) and we can speak of
$\Siiv(\bar h(\pmb x,\pmb y))_<$.   Since

$$\eqalign{\pmb x\restr\Cal S_n&
=\pmb x_{\star}\restr\Cal S_n
   +\Siiv(\bar h(\pmb x\restr\Cal S_n,
      \pmb y\restr\Cal S_n))_<\cr&
=\pmb x_{\star}\restr\Cal S_n
   +(\Siiv(\bar h(\pmb x,\pmb y))\restr\Cal S_n)_<
=\pmb x_{\star}\restr\Cal S_n
   +\Siiv(\bar h(\pmb x,\pmb y))_<\restr\Cal S_n\cr}$$

\noindent for each $n$,

\Centerline{$\pmb x=\pmb x_{\star}+\Siiv(\bar h(\pmb x,\pmb y))_<$.}

\medskip

{\bf (d)} If $\pmb x'$ is another S-integrable process such that
$\pmb x'=\pmb x_{\star}+\Siiv(\bar h(\pmb x',\pmb y))_<$, then
we see that

\Centerline{$\pmb x'\restr\Cal S_n
=\pmb x_{\star}\restr\Cal S_n
  +\Sii_{\pmb v\restr\Cal S_n}(\bar h(\pmb x'\restr\Cal S_n,
     \pmb y\restr\Cal S_n))_<$}

\noindent for each $n$.  Applying 654Mb in the reverse direction
to $\tilde{\Cal S}_n=\Cal S_{n+1}\vee\tau_n$, we see by induction that
$\pmb x'\restr\tilde{\Cal S}_n$ satisfies the same equation as
$\tildepmb x_n$ for each $n$.   By the uniqueness guaranteed in
654L, $\pmb x'\restr\tilde{\Cal S}_n=\tildepmb x_n$ for each $n$, and
$\pmb x'=\pmb x$.   Thus we have both existence and uniqueness in the case in
which $\Cal S$ has a greatest member.

\medskip

{\bf (e)} For the general case, as in part (f) of the proof of 654I, we
have a covering ideal $\Cal S'$ of $\Cal S$ such that $\pmb w\restr\Cal S'$
is a martingale.   For
each $\tau\in\Cal S'$ we have a partial solution $\pmb x_{\tau}$ with domain
$\Cal S\wedge\tau$.   These have a common extension to
a process $\pmb x$ with domain $\Cal S$, which will be the solution we seek
on the whole lattice $\Cal S$.
}%end of proof of 654N

\exercises{\leader{654X}{Basic exercises (a)}
%\spheader 654Xa
In 654I, show that if $\pmb u_{\star}$, $\pmb y$ and $\pmb v$ are
all near-simple then $\pmb u$ will be near-simple.    \Hint{apply 631Mb
to $\Cal S\subseteq[\min\Cal S,\sup\Cal S]$.}
%654I

\spheader 654Xb In 654I, suppose that $\pmb u'_{\star}$, $\pmb y'$ and
$\pmb u'$ are further
locally near-simple processes with domain $\Cal S$ such that
$\pmb u'=\pmb u'_{\star}+\iiv(\bar h(\pmb u',\pmb y'))$.   Show that
$\Bvalue{\pmb u'\ne\pmb u}\Bsubseteq\Bvalue{\pmb u'_{\star}\ne\pmb u_{\star}}
\Bcup\Bvalue{\pmb y'\ne\pmb y}$.
%654I and what about adjusting \pmb v?
%we'd need to adjust \pmb w, \hatpmb w

\spheader 654Xc Suppose that $\family{t}{T}{\frak A_t}$ is
right-continuous.   Let $\Cal S$ be an order-convex sublattice of
$\Cal T$ with a least member.   Suppose that $h:\BbbR^2\to\Bbb R$ is
a continuous function such
that $|h(\alpha,\beta)-h(\alpha',\beta)|
\le|\beta||\alpha-\alpha'|$ for $\alpha$, $\alpha'$, $\beta\in\Bbb R$.
Show that for any $\pmb u_{\star}$, $\pmb y\in\Mns(\Cal S)$ and any
integrator $\pmb v\in\Mns(\Cal S)$ there is a unique
$\pmb u\in\Mns(\Cal S)$ such that
$\pmb u=\pmb u_{\star}+\iiv(\bar h(\pmb u,\pmb y))$.   \Hint{for $K\ge 0$ set
$\pmb y_K=\med(-K\tbf 1^{(\Cal S)},\pmb y,K\tbf 1^{(\Cal S)})$.}
%654I 654Xb

\spheader 654Xd In 654I, show that if $\pmb v$ and $\pmb u_{\star}$
are locally jump-free then $\pmb u$ is locally jump-free.
%654I

\spheader 654Xe Show how 654I can be deduced from 654N.
%654N 646Qb

\spheader 654Xf Suppose that $\family{t}{T}{\frak A_t}$ is
right-continuous.   Let $\Cal S$ be an order-convex sublattice of
$\Cal T$ with a least member.   Suppose that $k\ge 1$, that
$h:\BbbR^{k+1}\to\Bbb R$ is
a locally bounded Borel measurable function
and that $K\ge 0$ is such that
$|h(\alpha,\beta_1,\ldots,\beta_k)
  -h(\alpha',\beta_1,\ldots,\beta_k))|\le K|\alpha-\alpha'|$ for all
$\alpha$, $\alpha'$, $\beta_1,\ldots,\beta_k\in\Bbb R$.
Let $\pmb v$ be a locally near-simple
local integrator with domain $\Cal S$.
(i) Show that if $h$ is continuous and
$\pmb u_{\star}$, $\pmb y_1,\ldots,\pmb y_k$ are locally near-simple
processes with domain $\Cal S$ there is a unique
locally near-simple process $\pmb u$ with domain $\Cal S$ such that
$\pmb u=\pmb u_{\star}+\iiv(\bar h(\pmb u,\pmb y_1,\ldots,\pmb y_k))$.
(ii) Show that if $\pmb x_{\star}$, $\pmb y_1,\ldots,\pmb y_k$
are locally S-integrable processes with domain $\Cal S$ there is a unique
locally S-integrable process $\pmb x$ with domain $\Cal S$ such that
$\pmb x=\pmb x_{\star}+\Siiv(\bar h(\pmb x,\pmb y_1,\ldots,\pmb y_k))_<$.
(iii) Show that if $\pmb u_{\star}$
is a locally near-simple process with domain
$\Cal S$ and $\pmb y_1,\ldots,\pmb y_k$ are locally S-integrable processes
with domain $\Cal S$ there is a unique locally near-simple process $\pmb u$
with domain $\Cal S$ such that
$\pmb u=\pmb u_{\star}+\Siiv(\bar h(\pmb u_<,\pmb y_1,\ldots,\pmb y_k))$.
%654N

\spheader 654Xg In 654N, show that if $\pmb v$ is
locally jump-free and $\pmb x_{\star}$
is locally near-simple then $\pmb x$ is locally jump-free.
%654N 641N 646Qb

\leader{654Y}{Further exercises (a)}
%\spheader 654Ya
Suppose that $\family{t}{T}{\frak A_t}$ is
right-continuous, and
that $\Cal S$ is an order-convex sublattice of $\Cal T$ with a least
element.   Write $\Mns$ for $\Mns(\Cal S)$.
Let $H_1$, $H_2:\Mns\to\Mns$ and $K\ge 0$ be such that, for both $j$,
($\alpha$) if $\pmb u$, $\pmb u'\in\Mns$, $\sigma\in\Cal S$ and
$\pmb u\restr\Cal S\wedge\sigma=\pmb u'\restr\Cal S\wedge\sigma$ then
$H_j(\pmb u)\restr\Cal S\wedge\sigma=H_j(\pmb u')\restr\Cal S\wedge\sigma$
($\beta$)
$\sup|H_j(\pmb u)-H_j(\pmb u')|\le K\sup|\pmb u-\pmb u'|$ for all $\pmb u$,
$\pmb u'\in\Mns$.   Show that for any near-simple integrator $\pmb v$ with
domain $\Cal S$ and any $\pmb u_{\star}\in\Mns(\Cal S)$ there is a
unique
$\pmb u\in\Mns$ such that $\pmb u=\pmb u_{\star}+H_1(\iiv(H_2(\pmb u)))$.
%654G
}%end of exercises


\endnotes{
\Notesheader{654} Rather against the general tendency of this treatise, I
have given a substantial amount of space to an independent proof of a theorem
(654I) which can be regarded as a corollary of a later result in the same
section (654N, 654Xe).   We could save a little paper
by going straight to 654N.   But, for once, I have tried to make the path
gentler, though longer, by introducing those ideas which can be used in the
context of the Riemann-sum integral, before tackling the extra technical
difficulties of the S-integral.

There is a switch between the equations
$\pmb u=\pmb u_{\star}+\iiv(\bar h(\pmb u,\pmb y))$ and
$\pmb x=\pmb x_{\star}+\Siiv(\bar h(\pmb x,\pmb y))_<$.
In the first, we are
dealing throughout with (locally) moderately oscillatory processes.
In the second, while $\pmb v$ remains a locally near-simple local
integrator, the other variables are locally S-integrable.   So we need a
locally S-integrable indefinite integral, and the definition in \S646 led
to a locally near-simple indefinite S-integral, just as the definition in
\S613 led to a locally near-simple indefinite Riemann-sum integral.
I have therefore turned to what one might call a previsible indefinite
S-integral.   (There is an immediate suggestion that we could look at
previsible definite S-integrals, but I shall not pursue this here.)
A re-shuffling of the notion is in 654Xf(iii).

The idea of the proof here is to follow the standard method in elementary
courses on ordinary
differential equations, using a contraction mapping on a suitable metric
space.   The trick is to find an appropriate metric.   In 654G I show that
the norm $\pmb u\mapsto\|\sup|\pmb u|\|_2$ will serve in a special case
which turns out to be adequate.   In fact I think that the F-norm
$j$ of that lemma is closer to the heart of the matter, as its formula
carries a hint of the way in which we have to treat martingales and
processes of bounded variation differently.

Of course the method can be adapted to go a great deal farther.   I think
that we can reach 654Ya without adding any substantial new ideas to those
in the proof of 654I.   I note also that the Lipschitz condition can be
significantly relaxed (654Xc, and see {\smc Protter 05}, \S V.3).
}%end of notes

\discrpage

% By 623O, $ii_{\hatpmb z}(\hatpmb z)$ is a local martingale
% so  $\hatpmb z^*-\hatpmb z^2$ is a local martingale
% also $\hatpmb z^*$ is uniformly integrable because
% $\hat z^*_{\max\Cal S}\in L^1$
% is $\hatpmb z^2$ uniformly integrable?

\frfilename{mt655.tex}
\versiondate{13.2.21}
\copyrightdate{2014}

\def\chaptername{Applications}
\def\sectionname{The Black-Scholes model}

\newsection{655}

This volume is supposed to be an introduction to stochastic integration for
a mathematically sophisticated but otherwise untutored readership.   You
will find it difficult to persuade anyone else to take your efforts
seriously if you do not have something to say
about its most famous applications,
starting with {\smc Black \& Scholes 73}.   I will therefore take the
space to give a very short account of the simplest model derived by their
method, even though the mathematical content is no more than simple
quotes from the work so far, and all the interesting ideas relate to
the theory of financial markets.

\leader{655A}{Stochastic differential equations} In \S\S651 and 653,
I expressed every result in terms of integral equations;  so that
Theorem 651B, for instance, reads

\Centerline{$\int_{\Cal S}\pmb u\,d\pmb z
=\int_{\Cal S}\pmb u\times\pmb z\,d\pmb v$.}

\noindent But the mnemonic for it, in the style of \S617, would be

\Centerline{$d\pmb z\sim\pmb z\,d\pmb v$,}

\noindent and some authors are happy to express this in the form
$\Bover{d\pmb z}{d\pmb v}=\pmb z$.   Similarly, where in Theorem 654I I
write

\Centerline{$\pmb u=\pmb u_{\star}+\iiv(\bar h(\pmb u,\pmb y))$,}

\noindent others might write

\Centerline{$u_{\min\Cal S}=u_{\star,\min\Cal S}$,
\quad$d\pmb u\sim d\pmb u_{\star}+\bar h(\pmb u,\pmb y)d\pmb v$}

\noindent or perhaps

\Centerline{$u_{\min\Cal S}=u_{\star,\min\Cal S}$,
\quad$\Bover{d\pmb u}{d\pmb v}=\Bover{d\pmb u_{\star}}{d\pmb v}
   +\bar h(\pmb u,\pmb y)$.}

\cmmnt{\noindent I myself prefer to regard these alternative expressions as
potentially suggestive abbreviations.   For instance, they erase the
distinction between the Riemann-sum integral and the S-integral;  as we
have seen in 654N, this can be a helpful stimulus, but does not release us
from the obligation to find sufficient conditions on the inputs
$\pmb v$, $\pmb x_{\star}$, $\pmb y$ and $h$ there.
}

\leader{655B}{A model of stock prices}
For the rest of this section, I will suppose that \Asissixt\ is a
right-continuous real-time stochastic integration structure
and $\Cal S$ is a non-empty ideal of $\Cal T$\cmmnt{, that is, an
order-convex sublattice with least member $\check 0=\min\Cal T$}.
Let $\pmb{\iota}$ be the identity process\cmmnt{ (612F)} and
$\pmb w$ a Brownian-type process on $\Cal S$\cmmnt{, that is,
a locally jump-free local martingale with domain $\Cal S$ and
quadratic variation $\pmb w^*=\pmb\iota\restr\Cal S$ (653K)}.
Consider the differential equation

\Centerline{$d\pmb u\sim\alpha\pmb ud\pmb{\iota}+\beta\pmb ud\pmb{w}$,
\quad$u_{\check 0}=u_{\star}$}

\noindent or

\Centerline{$\pmb u
=u_{\star}\tbf1 + \alpha ii_{\pmb\iota}(\pmb u)+\beta ii_{\pmb w}(\pmb u)
=u_{\star}\tbf1 + ii_{\tildepmb w}(\pmb u)$}

\noindent where $\tildepmb w=\alpha\pmb\iota+\beta\pmb w$ and
$u_{\star}\in L^0(\frak A_0)$.
Then $\tildepmb w$ is a locally jump-free local integrator and
its quadratic variation $\tildepmb w^*$ is \dvro{$\beta^2\pmb\iota$.}{

\Centerline{$\covar{\tildepmb w}{\tildepmb w}
\cmmnt{\mskip5mu=\alpha^2\covar{\pmb\iota}{\pmb\iota}
   +2\alpha\beta\covar{\pmb\iota}{\pmb w}
   +\beta^2\covar{\pmb w}{\pmb w}}
=\beta^2\pmb\iota$}

\noindent\cmmnt{by 624C, because $\pmb\iota$ is locally of bounded variation and
both $\pmb\iota$ and $\pmb w$ are locally jump-free.}}   So
the equation has solution

\Centerline{$\pmb u
=u_{\star}\,\overline{\exp}
   (\tilde{\pmb{w}}-\bover12\beta^2\pmb{\iota})$\dvro{,}{}}

\noindent\cmmnt{(put 651B and 613L(b-ii) together, as in 651Xa), }which
is a locally jump-free local integrator and
is unique\cmmnt{ (654I, with $h(x,y)=x$ and $\pmb u_{\star}=u_{\star}\tbf1$)}.

\leader{655C}{A model for options} Now suppose that we have an `option'
in a `stock' whose value is accurately modelled by the process $\pmb u$.
Our objective is to find a rational approach
leading to a way of determining the value $\pmb v$ of this option.
We assume that\cmmnt{ $\pmb v$ depends on the time
and the value of $\pmb u$
at that time;  that is, that} there is a function $h$ such that
$\pmb{v}=\bar h(\pmb u,\pmb{\iota})$.
(If $\pmb u$ and $\pmb v$ are
represented\cmmnt{, in the manner of 631D,} by real-valued
processes $\langle U_t\rangle_{t\ge 0}$ and
$\langle V_t\rangle_{t\ge 0}$ with \cadlag\ sample paths,
we are supposing that
$V_t(\omega)=h(U_t(\omega),t)$ for most pairs $(\omega,t)$.)
The terms of the option will
give us some information about the function $h$;  for instance, a call
option, allowing us to buy a quantity $c$ of the stock for a
strike price $x_1$ at expiry time $t_1$,
will then have value $h(x,t_1)=c\max(x-x_1,0)$,
because we shall be able to buy the stock at price $x_1$ and sell it at
price $x$;  if $x\le x_1$, we just do nothing.

We assume\cmmnt{ also} that $h$ is twice
continuously differentiable, with partial derivatives $h_1$, $h_2$ and
second partial derivatives $h_{11},\ldots,h_{22}$.   Then\cmmnt{ we have

$$\eqalign{\pmb{v}&
=\bar h(\pmb u,\pmb{\iota})
=\bar h(u_{\star},0) + ii_{\bar h(\pmb u,\pmb\iota)}(\tbf 1)\cr&
=\bar h(u_{\star},0)+ii_{\pmb u}(\bar h_1(\pmb u,\pmb{\iota}))
   +ii_{\pmb\iota}(\bar h_2(\pmb u,\pmb{\iota}))
   +\Bover12 ii_{\pmb u^*}(\bar h_{11}(\pmb u,\pmb{\iota}))
\cr}$$

\noindent by 619K, because $\pmb u$ and $\pmb{\iota}$ are jump-free local
integrators, and $\pmb{\iota}^*=\covar{\pmb{\iota}}{\pmb u}=\tbf0$.  In
the differential form, this becomes

\Centerline{$d\pmb v\sim\bar h_1(\pmb u,\pmb{\iota})d\pmb u
   +\bar h_2(\pmb u,\pmb{\iota})d\pmb\iota
   +\Bover12\bar h_{11}(\pmb u,\pmb{\iota})d\pmb u^*$.}

\noindent
To get an expression for the term $ii_{\pmb u^*}(\ldots)$, observe that
$\pmb u^*=(ii_{\tildepmb w}(\pmb u))^*$ (617Kc) and we can use
617Q(a-iii) to see that

\Centerline{$ii_{\pmb u^*}(\bar h_{11}(\pmb u,\pmb{\iota}))
=ii_{\tildepmb w^*}(\bar h_{11}(\pmb u,\pmb{\iota})\times\pmb u^2)
=\beta^2ii_{\pmb\iota}(\bar h_{11}(\pmb u,\pmb{\iota})\times\pmb u^2)$.}

\noindent So we have
}%end of cmmnt

\Centerline{$\pmb v
=\bar h(u_{\star},0)+ii_{\pmb u}(\bar h_1(\pmb u,\pmb{\iota}))
   +ii_{\pmb\iota}(\bar h_2(\pmb u,\pmb{\iota})
   +\Bover12\beta^2\pmb u^2\times\bar h_{11}(\pmb u,\pmb{\iota}))
$\dvro{.}{}}

\cmmnt{\noindent or, using the mnemonic for 617Q in the form
$d\pmb u^*\sim\pmb u^2d\pmb w^*\sim\beta^2\pmb u^2d\pmb\iota$,

\Centerline{$d\pmb v\sim\bar h_1(\pmb u,\pmb{\iota})d\pmb u
   +(\bar h_2(\pmb u,\pmb{\iota})
   +\Bover12\beta^2\pmb u^2\times\bar h_{11}(\pmb u,\pmb{\iota}))
      d\pmb\iota$.}
}%end of cmmnt

\leader{655D}{Hedging and a risk-free portfolio}
Still supposing that there is such a function $h$,
consider a hedged version of the option.   In addition to the option,
we `hedge' by a suitably varying
quantity $\bar g(\pmb u,\pmb{\iota})$ in the stock $\pmb u$
to give us a portfolio
$\tilde{\pmb{v}}=\pmb{v}-ii_{\pmb u}(\bar g(\pmb u,\pmb{\iota}))$ in
which the\cmmnt{ (so far unknown)} value of $\pmb v$ is modified by the
accumulated losses and gains of our hedging strategy.
\cmmnt{Let me try to explain the intuition behind this.
Actually I should perhaps begin with an explanation of the minus sign.}
The idea of a `hedge' is that we can `sell the stock short', that is,
sell stock we don't necessarily
possess;  \cmmnt{in a strictly regulated market, this will be
to some extent controlled by a requirement that we should borrow some stock
from a legitimate holder, and then sell that;  but in any case,} we take
cash now, and promise to buy the stock back soon {\it at the price then
ruling}\cmmnt{, either because we have to return it to its original owner, or
because we never delivered it}.
This is not an option, it is a contract.   From the point of
view of our counterparty, it is just like buying real stock.
\cmmnt{Of course they have to
trust us, but it is part of the theory of `perfect markets' that the
agents do trust each other.   There is a question about who decides the
buy-back time, but our counterparty doesn't much mind, because they will
be able to buy the stock in the market whenever we ask
(remember, we shall be paying the price at that time, and this is a
perfect
market, so there will always be buyers and sellers at that price).}
The idea behind this is that if we possess a call option, we stand to make
money if the stock rises in value, and not if it falls;
by going short, we hedge our bet to make our prospects more even.

We allow $g$ to take negative values;  this is because we
can `go long', that is, buy some stock with the intention of selling it
again if our strategy calls on us to do so.
Note that we believe that $\pmb u$
is jump-free, so can imagine adjusting the hedge
rapidly compared with changes in $\pmb u$.

We shall have

\Centerline{$\tilde{\pmb v}
=\bar h(u_{\star},0)
   +ii_{\pmb u}(\bar h_1(\pmb u,\pmb{\iota})-\bar g(\pmb u,\pmb\iota))
  +ii_{\pmb\iota}(\bar h_2(\pmb u,\pmb{\iota})
   +\Bover12\beta^2\pmb u^2\times\bar h_{11}(\pmb u,\pmb{\iota}))
$\dvro{.}{}}

\cmmnt{\noindent or

\Centerline{$\tilde v_{\check 0}=\bar h(u_{\star},0)$,
\quad$d\tilde{\pmb v}
\sim(\bar h_1(\pmb u,\pmb\iota)-\bar g(\pmb u,\pmb\iota))d\pmb u
 +(\bar h_2(\pmb u,\pmb{\iota})
  +\Bover12\beta^2\pmb u^2
      \times\bar h_{11}(\pmb u,\pmb{\iota}))d\pmb\iota$.}
}

\noindent So if we set $g=h_1$, we get

\Centerline{$\tilde{\pmb v}
=\bar h(u_{\star},0)
  +ii_{\pmb\iota}(\bar h_2(\pmb u,\pmb{\iota})
   +\Bover12\beta^2\pmb u^2\times\bar h_{11}(\pmb u,\pmb{\iota}))$,}

\Centerline{$d\tilde{\pmb v}
\sim(\bar h_2(\pmb u,\pmb{\iota})
   +\Bover12\beta^2\pmb u^2\times\bar h_{11}(\pmb u,\pmb{\iota}))
   d\pmb\iota$.}

\noindent Now this is `risk-free';  for a short time interval
$[\sigma,\sigma']$,

\Centerline{$\tilde v_{\sigma'}-\tilde v_{\sigma}
\bumpeq(\bar h_2(u_{\sigma},\sigma)
           +\Bover12\beta^2\bar h_{11}(u_{\sigma},\sigma))
   \times(\sigma'-\sigma)$}

\noindent is well approximated by something calculable from knowledge
of the stopping times $\sigma$, $\sigma'$ and the
situation at the starting time $\sigma$,
but not requiring any foreknowledge of the evolution of
$\pmb u$ or $\pmb w$.\cmmnt{   Consequently we can compare it with other
risk-free investments.}
Suppose that we can be sure of being able to borrow, or lend,
cash, at an interest rate $\rho$, with complete safety.   We
are supposed to be operating in a perfect market, in which every agent
knows just what we know about $\pmb w$ and $\pmb u$, and can do the same
calculations, so that the process
$\bar h(\pmb u,\pmb\iota)$ describes the evolution of
the market price, either buying or selling, of the option.
We therefore expect $\tilde v_{\sigma'}-\tilde v_{\sigma}$, the
agreed expected profit from holding the portfolio $\tilde v$ from
time $\sigma$ to time $\sigma'$, to be very close to
the expected income over that time period if we sell our option and
our holding in the stock $\pmb u$, and invest the net proceeds in
a bond at interest rate $\rho$.

At this point I need to remark that these net proceeds will not be the
current value $\tilde v_{\sigma}$.   The process $\tilde{\pmb v}$ takes
past gains and losses into account in the term
$ii_{\pmb u}(\bar g(\pmb u,\pmb\iota))$.\cmmnt{   But these are water
under the
bridge.}   Our holding at, and immediately after, the time $\sigma$ is
$v_{\sigma}-\bar g(u_{\sigma},\sigma)\times u_{\sigma}$, because if $g$ is
positive, that is, we are shorting the stock, we shall have to buy it back
at once to liquidate our position, while if $g$ is negative, that is, we
are holding some stock, we will sell it.   So we expect that

\Centerline{$\tilde v_{\sigma'}-\tilde v_{\sigma}
\bumpeq \rho(v_{\sigma}-\bar h_1(u_{\sigma},\sigma)\times u_{\sigma})
   \times(\sigma'-\sigma)$,}

\noindent that is,

\Centerline{$d\tilde{\pmb v}
\sim\rho(\pmb v-\bar h_1(\pmb u,\pmb\iota)\times\pmb u)d\pmb\iota$,}

\noindent and matching the two formulae for $d\tilde{\pmb v}$ we get

\Centerline{$\bar h_2(\pmb u,\pmb\iota)
  +\Bover12\beta^2\pmb u^2\times\bar h_{11}(\pmb u,\pmb\iota)
=\rho(\pmb v-\pmb u\times\bar h_1(\pmb u,\pmb\iota))
=\rho(\bar h(\pmb u,\pmb\iota)-\pmb u\times\bar h_1(\pmb u,\pmb\iota))$.}

\noindent To ensure this, we shall have to have

\Centerline{$h_2(x,t)+\Bover12\beta^2x^2h_{11}(x,t)
=\rho(h(x,t)-xh_1(x,t))$}

\noindent for all relevant $x$ and $t$, that is,

\Centerline{$\Pd{h}{t}+\Bover12\beta^2x^2\Pdd{h}{x}
   +\rho x\Pd{h}{x}-\rho h=0$}

\noindent which is the {\bf Black-Scholes equation} for the evolution of the
value $h(x,t)$ of an option in a stock with price $x$ at time $t$.



%\leader{655F}{}
%\leader{655G}{}
%\leader{655H}{}
%\leader{655I}{}
%\leader{655J}{}
%\leader{655K}{}
%\leader{655L}{}
%\leader{655M}{}
%\leader{655N}{}
%\leader{655O}{}
%\leader{655P}{}
%\leader{655Q}{}
%\leader{655R}{}
%\leader{655S}{}
%\leader{655T}{}
%\leader{655U}{}
%\leader{655V}{}
%\leader{655W}{}

\exercises{%\leader{655X}{Basic exercises (a)}

%\leader{655Y}{Further exercises (a)}

}%end of exercises


\endnotes{
\Notesheader{655} For obvious reasons, an enormous amount of work has been
done on adaptations and correction terms for the Black-Scholes equation.
But I will indulge myself by saying why I think that most of this effort
has been wasted, at least from the point of view of those who have paid for
it.   Up to the end of 655C we are
enjoying ourselves with some pretty mathematics, and the finance
theory of 655D, in the abstract, offers us some interesting
new challenges.   But look at the
`supposes' and `assumes' there.   Why should we believe that the
value of a stock can be described by the formulae of 655B?   Empirical
evidence from stock exchange records shows that many
stocks have followed this pattern for years at a time;  but companies do
quite often go bankrupt or get taken over,
and the model so far has no place for either.   Can we be
sure that the value of an option depends only on the time and on the
present value of the stock?   The argument I presented above pays no
attention to risk-aversion, because if the rest of the theory were sound,
there would be no risk;  but even hedge funds are, in practice,
risk-averse,
and only intermittently do they have blind faith in the theory.
Why should the
function $h$, if it exists, be twice continuously differentiable?
The model I have described assumes that the parameters $\alpha$, $\beta$
and $\rho$ are constant.   All of these
we can hope to do something about -- but
only if we have a convincing account of how they will change in the future.
There are ways of inferring what the majority of
currently active traders expect them
to do, but are we supposed to believe that `the market' is always right
about such things, when we read daily of gross blunders, and when interest
rates, in particular, are influenced so heavily by government action?
I wrote `we can imagine adjusting the hedge rapidly compared with changes
in $\pmb u$'.   A very large amount of money has been, and is being, spent
on speeding up trading processes to make this true.   But what if there is
enough trading in the options on a stock to make the corresponding hedges
a significant influence on the price of the stock?

And then there is all the stuff about perfect markets.
One of the greatest advantages
claimed for open markets is that prices give a rapid and trustworthy
way for information to
percolate through the system;  but in more than one financial market
in 2012-2014, the
volume of trading, the bulk of it between rival pricing models, offered
large rewards for fraudulent manipulation of price indices, which duly
occurred.   On most trading days,
to be sure, and for most stocks and many options,
there are active agents and (among themselves) good information and
low transaction costs;  but local market
failures, when trading in individual stocks is suspended, are common,
and more general failures, in which whole sectors are
briefly paralyzed, occur in most decades.

One of these failures is particularly associated with the Nobel
prizewinners Myron Scholes and Robert Merton, who belonged to the limited
partnership Long-Term Capital Management which came spectacularly to grief
in 1998.   For an account of the rise and fall of LTCM, see
{\smc Lowenstein 01}.   Their business model amounted to betting when the
odds seemed right.   Typically they were accepting other peoples' hedges,
acting as a kind of unregulated insurance company.
Of course they could do this only because {\it not}
everyone took exactly the same view of the options involved, and the margins
on the bets were necessarily narrow, so the success of the fund depended
on betting very heavily.   They reached the level at which their own
transactions were having a substantial effect, and at which quite
possibly temporary variations in market prices were leading to changes in
the value of their assets sufficient to alarm not only their investors but
their counterparties and creditors and the Federal Reserve Bank of New
York.   With hindsight we can see that they made a string of bad
decisions;  but a common feature of these was that they assumed too readily
that they could wait for anomalies to evaporate.   In particular, they
tried to take advantage of a reluctance on the part of other investors to
hold Russian bonds, both private and governmental.  This anomaly did
not evaporate, and culminated in defaults.
LTCM seem to have been using pricing
models which took inadequate account of such risks.

One would have hoped that the catastrophe would have led to more
realistic assessments by financiers.   But that didn't happen.
Lehman Brothers under Richard Fuld were acting for LTCM all through the
crisis of 1998.
In 2008 it was Fuld's turn to destroy his company by obstinately
hanging on through warning signals.   Nobody that year was imprudently
exposed to Russia.   They had found ways of being imprudently
exposed to mortgages.

To put it bluntly:  I do not think that this can be fixed by tinkering.
Option pricing began because people had good reasons for wanting to buy
options, and Black, Scholes, Merton and others offered formulae with
reasonably plausible justifications, at least in comparison with what had
gone before.   But the many mathematical
refinements which have been developed since have been
no substitute for good judgement.
The enormous structures which have been built on these
ideas in some countries are, I believe, cancerous growths,
and should be starved into remission before radical surgery becomes forced.
}%end of notes

\discrpage



\references{
\frfilename{mt6r.tex}
\versiondate{10.2.23}
\copyrightdate{2022}

% except for references in ...X., ...Y., ...Z. give locations in form
%  342L  rather than  342Lb
%  \S511 {\it notes\/}, \S512 {\it intro.}]


\gdef\topparagraph{}
\gdef\bottomparagraph{}
\gdef\newparagraph{}
\gdef\chaptername{References}
\gdef\sectionname{References}
\gdef\headlinesectionname{References}


\def\bsdot{.\penalty-100}

\centerline{\bf References for Volume 6}

\medskip%A\medskip%B

Bain A.\ [09] {\it Stochastic Calculus.}
{\tt http://www.chiark.greenend.org.uk/$^{\sim}$alanb/}, 25.5.09.
%bain.pdf

Bichteler K.\ [79] `Stochastic integrators',
Bull.\ Amer.\ Math.\ Soc.\ (2) 1 (1979) 761-765.
%http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&page=toc&handle=euclid.bams/1183544712
\cmmnt{[627I.]}

Bichteler K.\ [81] `Stochastic integration and $L^p$ theory of
martingales', Ann.\ Prob.\ 9 (1981) 49-89.
%https://projecteuclid.org/euclid.aop/1176994509
\cmmnt{[649C.]}

Black F.\ \& Scholes M.\ [73] `The pricing of options and corporate
liabilities', J.\ Political Economy 81 (1973) 637-659.
\cmmnt{[\S655 {\it intro.}]}

Burkholder D.L.\ [66] `Martingale Transforms',
Ann.\ of Math.\ Stat.\ 37 (1966) 1494-1504.
\cmmnt{[621H.]}
%burkholder.66.pdf

%\medskip%C
\medskip%D

\leaveitout{
Delbaen F.\ \& Schachermeyer W.\ [06] {\it The Mathematics of Arbitrage.}
Springer, 2006.
ISBN-13: 978-3540219927
HG 6024.A3D4  0620 6436 3
}

\leaveitout{
Dellacherie C.\ \& Meyer P.-A.\ [79]
Probabilities and Potential, Volume 1
North-Holland, 1979  (Mathematics Studies).
% ISBN: 072040701X chapters I-VI
}

Dellacherie C.\ \& Meyer P.-A.\ [82]
{\it Probabilities and Potential B. Theory of Martingales.}
Elsevier, 1982 (North-Holland Mathematics Studies 72).
%ISBN: 0444865268 chapters V-VIII
%translator J.P.Wilson
\cmmnt{[627I.]}

\leaveitout{
Dellacherie C.\ \& Meyer P.-A.\ [88]
{\it Probabilities and Potential C: Potential Theory for Discrete and
Continuous Semigroups.}
Elsevier, 1988
%ISBN: 0444703861 (pt.C)
}

\leaveitout{
Dellacherie C., Meyer P.-A.\ \& Weil M.\ [77] (eds.)
{\it S\'eminaire de Probabilit\'es XI.} Springer, 1977
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% ISBN 978-3-540-08145-6
}

\leaveitout{
Dol\'eans-Dade C.\ [76] `On the existence and unicity of solutions of
stochastic differential equations', Z.\ Wahrscheinlichkeitstheorie verw.\
Gebiete 36 (1976) 93-101.
%? fundamental theorem of martingales
% ILL 15.9.12
}

%\medskipE\medskip%F

\leaveitout{
Fremlin D.H.\ [n11] `Vector-valued gauge integrals', note of 12.7.11
({\tt http:{\bsp}/www1.essex.ac.uk{\bsp}maths{\bsp}people{\bsp}fremlin{\bsp}preprints.htm}).
%n11527
}

%\medskipG\medskip%H
\medskip%I

It\^o K.\ [44] `Stochastic Integral', Proc.\ Imperial Acad.\ Tokyo 20
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\cmmnt{[Vol.\ 6 {\it intro.}, Chap.\ 62 {\it intro.}]}

\leaveitout{
\medskip%J

Jacod J.\ [79] {\it Calcul Stochastique et Probl\`emes de Martingales.}
Springer, 1979 (Lecture Notes in Mathematics 714).
}

\medskip%K

Karandikar R.L.\ [95] `On pathwise stochastic integration', Stochastic
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\cmmnt{[649C.]}

Karatzas I.\ \& Shreve S.\ [91] {\it Brownian Motion and Stochastic
Calculus.}  Springer, 1991.
\cmmnt{[621J.]}

\leaveitout{
Kazamaki N.\ [77] `On a problem of Girsanov',
T\^ohoku Math.\ J.\ (2) 29 (1977) 597-600.
}

\leaveitout{
Kunita H.\ \& Watanabe S.\ [67] `On square integrable martingales', Nagoya
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}

\medskip%L

Lowenstein R.\ [01] {\it When Genius Failed: the rise and fall of
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%HG 4928.5,  0120 9001 8,  0520 6473 5
\cmmnt{[\S655 {\it notes\/}.]}

\leaveitout{
\medskip%M

Meyer P.A.\ [76] `Un cours sur les int\'egrales stochastiques',
pp.\ 245-400 in {\smc Dellacherie Meyer \& Weil 77}.
}

\medskip%N

\leaveitout{
Novikov A.A.\ [72] `On an identity for stochastic integrals',
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}

Nutz M.\ [p11]  `Pathwise construction of stochastic integrals',
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\cmmnt{[649E.]}

\medskip%O\medskip%P

Protter P.E.\ [05] {\it Stochastic Integration and Differential Equations.}
Springer, 2005.
\cmmnt{[\S627 {\it notes\/}, 634Yd, \S64 {\it intro.\/}, \S643 {\it notes\/}, 
\S645 {\it notes\/}, \S654 {\it notes\/}.]}
 % ISBN 978-3-540-00313-7 419 pp

\medskip%Q\medskip%R

Rogers L.C.J.\ \& Williams D.\ [00] {\it Diffusions, Markov Processes
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\medskip%S

Sato K.-I.\ [13] {\it L\'evy processes and infinitely divisible distributions.} 
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\cmmnt{[\S652 {\it notes\/}.]} 
% Cambridge studies in advanced mathematics 68. 

%Seifried F.\ {\tt seifried\@mathematik.uni-kl.de}
%\medskip%T\medskip%U\medskip%V

\leaveitout{
\medskip%W

Weizs\"acker H.von \& Winkler G,\ [90] {\it Stochastic Integrals.}
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}

\medskip%X\medskip%Y\medskip%Z

\discrpage

}%end of references
%https://almostsure.wordpress.com/2009/11/30/predictable-stopping-times/
%https://almostsure.wordpress.com/  George Lowther
%   george.lowther@blueyonder.co.uk   (2009)

%---------------------------------------------------------------%

\end



