Contents of Measure Theory, by D.H.Fremlin

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=∫Su 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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Revised 19.12.13