Contents of Measure Theory, by D.H.Fremlin

Chapter 62: Martingales

621 Finite martingales
Uniform integrability; martingales and submartingales; Doob's maximal inequality; decomposition of a submartingale into a martingale and a trend; SI(u,dv) for ∥ ∥-bounded u and martingales v.

622 Fully adapted martingales
Conditional expectations; martingales, local martingales and semimartingales; the martingales Pz; semimartingales are integrators.

623 Virtually local martingales
Operators RA; virtually local martingales; indefinite integrals.

624 Quadratic variation
Covariation of virtually local martingales; constant quadratic variations; the quadratic variation of Brownian motion; ∥ ∥2-bounded martingales.

625 Changing the measure
Radon-Nikodým derivatives and conditional expectations; semi-martingales remain semi-martingales under change of law.

626 Submartingales
Submartingales; finitely-covered envelopes; interval functions PΔv; previsible variations; the Doob-Meyer theorem; semi-martingales.

627 Integrators and semimartingales
Supermartingales, quasi-martingales and strong integrators; non-negative supermartingales; integrators are semi-martingales; changing law to make an integrator a strong integrator.

*628 Refining a martingale inequality
Interpolating in a finite martingale; associating a martingale with an L-bounded martingale; a better constant in a key inequality; the capped-stake variation set of a martingale; the quadratic variation of a martingale.

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Revised 11.5.2018