Chapter 63: Back to work
631
Bounded variation
|dv| and dv↑; near-simple and jump-free
processes of bounded variation; indefinite integrals; quadratic
variation; the jump-free part of a process of bounded variation.
632
Changing the measure
Radon-Nikodým derivatives and conditional expectations;
semimartingales stay semimartingales under change of law.
633
Martingales and submartingales
Martingales, local martingales and uniform integrability; interval
functions PΔv; submartingales; the Doob-Meyer theorem
and previsible variations.
634
Integrators and semimartingales
Supermartingales, quasi-martingales and strong integrators;
decomposition of quasi-martingales into differences of
supermartingales; convex sets in L1μ which are
topologically bounded in L0; changing law to make an integrator a
strong integrator (the Bichteler-Dellacherie theorem).
635
The ucp topology
The ucp topology on the space Mob of order-bounded processes;
continuity of basic operations; completeness; the spaces of
near-simple and jump-free processes; indefinite integration with
respect to a near-simple integrator is a continuous linear operator;
if both integrator and integrand are near-simple, the indefinite
integral is near-simple; if the integrand is near-simple and the
integrator is jump-free, the indefinite integral is jump-free;
change of variable in an integral ∫z dvdv'.
636
Coverings and full lattices
Covering sets and sublattices; extension of processes defined on
covering sublattices; properties of such extensions; the ucp
topology for full lattices; quadratic variations of jump-free
processes.
637
Separating sublattices
Separating sublattices; cases when
∫Su dv=∫S′u dv;
using a measure-converging filter for path-by-path calculation of a
stochastic integral; connections between properties of a process and
its restriction to a separating sublattice; a general extension
method for near-simple processes.
638
Changing the algebra
Subalgebras and restricted filtrations and stopping time lattices;
fully adapted processes and absolute properties; càdlàg processes
and their extensions; relative independence and subalgebras
coordinated with filtrations; martingales; stochastic independence
and free products.
639
Changing the filtration
The filtration defined from a totally ordered family of stopping
times; the corresponding transformation of fully adapted processes;
integration by substitution.
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