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; *S _{I}*(

622
Fully adapted martingales

Conditional expectations; martingales, local martingales and
semimartingales; the martingales * Pz*; semimartingales are
integrators.

623
Virtually local martingales

Operators *R**A*; 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
*P*Δ* v*; 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