Quantitative Finance

1Stochastic Processes: A Brief Review

1.1 Introduction

In this chapter, we introduce the basic mathematical tools we will use. We assume the reader has a good understanding of probability spaces and random variables. For more details we refer to [67, 70]. This chapter is not meant to be a replacement for a book. To get the fundamentals please consult [70, 117]. In this chapter, we are reviewing fundamental notions for the rest of the book.

So, what is a stochastic process? When asked this question, R.A. Fisher famously replied, “What is a stochastic process? Oh, it's just one darn thing after another.” We hope to elaborate on Fisher's reply in this introduction.

We start the study of stochastic processes by presenting some commonly assumed properties and characteristics. Generally, these characteristics simplify analysis of stochastic processes. However, a stochastic process with these properties will have simplified dynamics, and the resulting models may not be complex enough to model real‐life behavior. In Section 1.6 of this chapter, we introduce the simplest stochastic processes: the coin toss process (also known as the Bernoulli process) which produces the simple random walk.

We start with the definition of a stochastic process.

Definition 1.1.1

Given a probability space , a stochastic process is any collection of random variables defined on this probability space, ...