CHAPTER 8 - An Introduction to Stochastic Processes and, in Particular, Brownian Motion

Abstract

A stochastic process is a collection of random variables, {Xt, t □ T}, all defined over the same probability space, where T is an indexing set and Xt is a random variable for each t. This chapter discusses the foundation for the theory of stochastic processes and the nature of stochastic processes and focuses on a particular stochastic process known as separable Brownian motion. Two problems are encountered in the foundations of the theory of stochastic processes. One problem is that of defining a probability of a set of elementary events obtained by an uncountable intersection or union of events that one might wish to be an event. A second problem encountered in the theory of stochastic processes is in fitting the process over its proper function space. The problem arises concerning conditions under which a particular function space could serve as a probability space for a certain stochastic process. Both the problems are solved by means of the notion of separability of a stochastic process. The chapter explains the separability of stochastic processes and presents a proof of the fundamental theorem about separability.