CHAPTER 7 - Conditional Expectation and Martingale Theory

Abstract

Conditional expectation is one of the fundamental notions in probability theory and is a most frequently used tool. This chapter discusses the conditional expectation and some of its fundamental properties. The only problem connected with conditional form of the Lebesgue dominated convergence theorem is that of obtaining a conditional form of the monotone convergence theorem. One of the applications of the martingale convergence theorem is in Brownian motion. This application is a notable theorem because of Paul Levy's construction on Brownian motion. The chapter also discusses the martingale and submartingale convergence theorems, the properties, and the application of martingale theory. In the subsequent treatment of martingales and submartingales, a conditional form of Jensen's inequality is needed. Convex functions possess certain properties and one such property is that every convex function is continuous.