Martingales

Abstract

The definitions, simplest properties and first examples of martingales and sub/super-martingales are given in Sect. 15.1. Stopping (Markov) times are introduced in Sect. 15.2, which also contains Doob’s theorem on random change of time and Wald’s identity together with a number of its applications to boundary crossing problems and elsewhere. This is followed by Sect. 15.3 presenting fundamental martingale inequalities, including Doob’s inequality with a number of its consequences, and an inequality for the number of strip crossings. Section 15.4 begins with Doob’s martingale convergence theorem and also presents Lévy’s theorem and an application to branching processes. Section 15.5 derives several important inequalities for the moments of stochastic sequences.