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Stochastic Differential Equations: An Introduction with Applications

Bernt Oksendal

No book review section would be complete without at least a paragraph on this book. Oksendal has become a standard text for most courses in financial maths and stochastic calculus at postgraduate/final year undergraduate level. This is intended to be an introduction to the subject but readers need to be weary. This is far from an easy read. The book is written by a mathematician, for the mathematician and some basic knowledge of measure theory is assumed. All the material and theorems are introduced very rigorously and you may at times find yourself lost amidst all the theory.

Chapter 1 nicely introduces the motivation behind studying stochastic calculus. Chapter 2 is the best introduction to probability theory that I have ever seen – short and to the point. Chapter 3 introduces the backbone of stochastic calculus: the Ito integral, and the going starts to get tough. The author pays a lot of detail to the construction of the integral which will only benefit readers wish to understand the subject from an ‘analysis’ perspective. The next 5 chapters treat the core of stochastic calculus (SDEs, diffusions, filtering and changes of measure). All the concepts are treated extensively with in depth rigour. Again, unless you are looking at this text for a pure maths course you are unlikely to benefit from the rigour and it may in fact complicate things unnecessarily. The latest edition contains a chapter dedicated to mathematical finance. I personally feel that this is a very good addition to the book as it helps in understanding the intuition behind option pricing theory. But as I have already emphasised, readers need solid grounding in analysis and need to hold their nerve when confronted with rigorous theorems and proofs in order to benefit from the chapter.

Although the material may be difficult to read the problems are an excellent source of learning. Many of the questions are ‘applied’ problems which is nice considering the ‘pure maths’ nature of the text. Even if it is just for chapter 2 and the problems at the end of each section, I would recommend students of quantitative finance to pick up this book at some stage.

Reviewed by Samy Mohammed. Click here to buy this book at Amazon.co.uk.

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