Martingales

Abstract

A martingale is a sequence of discrete random variables indexed by a time parameter with the property that the conditional expectation of a future term given the past and present terms is the present term. It can be thought of as a stochastic model for the gambler’s fortune in a fair game. In Section 3.1 we formalize and generalize this definition and motivate it with examples. In Section 3.2 we prove the optional stopping theorem for martingales, which will play an important role in what follows. In Section 3.3 we prove the martingale convergence theorem.