Probability and Stochastic Processes

Chapter 16Stochastic Differential Equations with respect to Brownian Motion

This is a chapter not normally included in a traditional textbook on stochastic processes. However, it is a crucial chapter for any and all applications of stochastic processes to finance. Since my main area of application is finance, this chapter is absolutely necessary for my students.

I am going to start this chapter with a motivating argument. Consider a regular differential equation

where b is just a function of two variables. For example, will produce a linear ordinary differential equation (ODE)

or the way it is commonly written in ODE textbooks

These types of equations were good 50 or 100 years ago when we could not really observe things and when the models used were very simple. However, today we are dealing with, can measure, and can observe a very dynamic world. From these observations, we know that we need to construct better models than in a deterministic world. To cope with the random nature of things, at the beginning of the last century Albert Einstein and others postulated principles that ...