Chapter 3 Martingales and random walks

Abstract

This chapter presents the modern and general discrete time definitions of martingale, supermartingale, and submartingale. The chapter outlines three of the main areas of martingale results: martingale transforms, including transformations under systems of optional stopping, martingale inequalities, and the martingale convergence theorems. It describes some sample applications of these results in probability, statistics, and management science. The optional stopping theorems find direct application in management science to decision problems of an optimal stopping format. For continuous time martingales and stochastic integration, many important topics and applications are highly technical and can be properly stated only after many definitions are made and a new vocabulary is introduced. The chapter provides an introduction to this vocabulary beginning. A heuristic principle of stochastic modeling asserts that every process can be viewed as a Markov process if enough history is included in the state description and the modern theory of stochastic integration, in which martingale theory is basic, provides a framework for carrying out this program.