Chapter 63: Back to work

631
Bounded variation

|*d*** v**| and

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 *P*Δ* v*; 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 *L*^{1}_{μ} which are
topologically bounded in *L*^{0}; changing law to make an integrator a
strong integrator (the Bichteler-Dellacherie theorem).

635
The ucp topology

The ucp topology on the space *M*_{ob} 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* d

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
∫_{S}*u**d*** v**=∫

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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Revised 19.12.13