Probability and Stochastic Processes

Chapter 9Introduction to Stochastic Processes

What is a stochastic process?

Definition 9.1

Given a probability space , a stochastic process is any collection of random variables defined on this probability space. More specifically, the collection of random variables or alternatively written , where is the index set. We will alternate between the notations and to denote the value of the stochastic process at time t.

We give here the famous R.A. Fisher quote, who answered the same question:

What is a stochastic process? Oh, it's just one darn thing after another.

In this chapter, we start the study of stochastic processes by presenting common properties and characteristics. Generally, possessing one or more of these properties will simplify the study of the particular stochastic process. Not all stochastic processes have these properties. In the second section of this chapter, ...