Random Walks and Stochastic Differential Equations

Abstract

A stochastic process is a pair {x,t ≥ 0} or also written x (t) with f (x,t) denoting the probability distribution that the process time path assumes a real value x at time t. The study of stochastic processes has its origin in the study of kinetic behaviour of molecules in gas by physicists in the 19th century. It is only in this century, following works by Einstein, Kolmogorov, Levy, Wiener and others that stochastic processes have been studied in some depth. Bachelier, already in his dissertation in 1900 provided a study of stock exchange speculation establishing a connection between price fluctuations in the stock exchange and Brownian motion, an important class of stochastic processes to be studied here. In addition, Bachelier constructed the mathematical model for a fair game, or the martingale as we saw in the appendix in Chapter 2 and as we shall see in Chapter 4.