Quantitative Finance

9Stochastic Differential Equations

9.1 Introduction

In Section 1.8, we briefly defined a stochastic differential equation (SDE) and presented some examples. In this chapter, we will discuss a more intuitive way to arise to an SDE. We will begin by discussing the construction of a stochastic integral and an SDE and we will present some properties. Methods for approximating stochastic differential equation (SDEs) will also be discussed. This chapter is very crucial for many applications of stochastic processes to finance. SDEs have been briefly discussed in Chapter 1 of this textbook. However in order to make this chapter self contained, we will repeat some of the concepts earlier introduced and then discuss other concepts that will be used throughout this book.

We recall in Chapter 1 of this book that an SDE is a deterministic differential equation which is perturbed by random noise. Suppose we define an equation of the form:

The term “noise” needs to be properly defined. Assuming we define a stochastic process such that “noise”. Then we have:

(9.1)

This process was named the “white ...