Introduction

Observations of prices of stocks, positions of a diffusing particle and many other processes observed in time are often modelled by a stochastic process. A stochastic process is an umbrella term for any collection of random variables { X( t)} depending on time t. Time can be discrete, for example, t = 0,1,2, , or continuous, t ? 0. Calculus is suited more to continuous time processes. At any time t, the observation is described by a random variable which we denote by X _t or X( t). A stochastic process { X( t)} is frequently denoted by X or with a slight abuse of notation also by X( t).

In practice, we typically observe only a single realization of this process, a single path, out of a multitude of possible paths. Any single path is a function of time t, x _t = x( t), 0 ? t ? T; and the process can also be seen as a random function. To describe the distribution and to be able to...