Problems and Solutions in Mathematical Finance: Stochastic Calculus, Volume I

Chapter 3Stochastic Differential Equations

A stochastic differential equation (SDE) is a differential equation in which one or more of the terms has a random component. Within the context of mathematical finance, SDEs are frequently used to model diverse phenomena such as stock prices, interest rates or volatilities to name but a few. Typically, SDEs have continuous paths with both random and non-random components and to drive the random component of the model they usually incorporate a Wiener process. To enrich the model further, other types of random fluctuations are also employed in conjunction with the Wiener process, such as the Poisson process when modelling discontinuous jumps. In this chapter we will concentrate solely on SDEs having only a Wiener process, whilst in Chapter 5 we will discuss SDEs incorporating both Poisson and Wiener processes.

3.1 Introduction

To begin with, a one-dimensional stochastic differential equation can be described as

where is a standard Wiener process, is defined as the drift and the volatility. Many financial models can be written in this form, ...