Stochastic Differential Equations

Among financial engineers, Oksendal is one of the most popular stochastic calculus texts. This is partly due to its pedigree. First published in 1985, it is now in its sixth edition—so it has had plenty of time to establish itself. It is more advanced than the competing classic by Malliaris and Brock (1982).

 

The book is well written. Although mathematically advanced—it assumes a strong background in measure theory—the author connects well with his audience. Amidst the theorems and proofs, he emphasizes how to actually use the mathematics. No other book delves into the practicalities of solving stochastic differential equations the way this one does. There are lots of exercises—and solutions are provided for many.

Oksendal opens with a unique perspective. While other authors approach stochastic calculus from a probabilistic standpoint that eventually leads to stochastic differential equations, Oksendal starts with ordinary differential equations. He considers some standard deterministic differential equations related to population growth, electric circuits, etc. He then asks: what if we introduce some random "noise" into the equations? How would that randomness affect our solution? This is a wonderful, intuitive way to introduce the subject.

Contents

1. Introduction

2. Some Mathematical Preliminaries

3. Ito Integrals

4. The Ito Formula and the Martingale Representation Theorem

5. Stochastic Differential Equations

6. The Filtering Problem

7. Diffusions: Basic Properties

8. Other Topics in Diffusion Theory

9. Applications to Boundary Value Problems

10. Application to Optimal Stopping

11. Application to Stochastic Control

12. Application to Mathematical Finance

A. Normal Random Variables

B. Conditional Expectation

C. Uniform Integrability and Martingale Convergence

D. An Approximation Result

Solutions and Additional Hints to Some of the Exercises

Another difference between Oksendal and many alternative texts is the fact that Oksendal does not specifically target a financial audience. Almost all topics covered are of some financial interest. For example, the Kalman filter isn't used in derivatives pricing, but it is used in time series analysis. However, the relative emphasis given to various subjects is not tailored to financial applications. With the fifth edition, a chapter on financial applications—and especially Black-Scholes theory—was appended to the back of the book. This is very much an afterthought.

I highly recommend this book for mathematically sophisticated readers who want to have a working—as opposed to merely theoretical—understanding of stochastic calculus. If you want to perform continuous-time financial research or actually solve stochastic differential equations, this is the book. In some respects, the fact that the book is not targeted to a financial audience is nice. This broader perspective communicates a greater appreciation for stochastic calculus overall. It is a technical book. Some prior familiarity with stochastic calculus is desirable but not essential.

 

 

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