Brownian Motion

Abstract

So far in this book, we have studied stochastic processes in discrete time and continuous time but always restricted the state space to be discrete and in most cases a finite set. This restriction was necessitated by technical difficulties that arise when dealing with continuous state space. In this chapter, we shall study one very special stochastic process in continuous time and continuous state space. It is called Brownian motion in honor of the biologist Brown, who observed (using a microscope) that small particles suspended in a liquid perform a very frenzied-looking motion. The process is also called the Wiener process in honor of the probabilist who provided the rigorous mathematical framework for its description.We shall see that the normal random variable plays an important role in the analysis of Brownian motion, analogous to the role played by exponential random variables in Poisson processes. Hence we start with the study of normal random variables first.