Martingales and More General Markov Processes

Abstract

A martingale is a stochastic process in which, conditionally on its history, every future value is expected to remain at its current level of the process. Such situations are, for example, seen in fair games in which both the gambler and the gambling house are expected to break even in the long run. Martingales, together with their extensions to sub- and supermartingales, appear all over in stochastic modeling, and they provide us with powerful tools and techniques for addressing questions such as convergence of stochastic processes, limiting distributions, and methods related to stopping times (optional stopping theorems).