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 and local martingales; the
martingales Pz; martingales are local integrators;
L2-martingales.
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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