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Stochastic Calculus and Financial Applications

North American Actuarial Journal, Apr 2002 by Young, Virginia R

Steele, J. Michael, 2000, Stochastic Calculus and Financial Applications, Springer-Verlag, New York, 315 pages, $69.95.

In this book, J. Michael Steele presents topics in stochastic calculus in the conversational style of a lecture. He begins with the simple random walk, then progresses quickly to Brownian motion, martingales, Ito's formula, stochastic differential equations, Girsanov theory, and the Feynman-Kac formula.

The portion of the book covering financial applications is a bit skimpy. However, if one wants a solid background in the mathematics of stochastic calculus, then this is an excellent book. After working through Steele's book, good ones to study to learn more about applications in finance are: Arbitrage Theory in Continuous Time, by Tomas Bjork (Oxford University Press, 1998) and Introduction to Stochastic Calculus Applied to Finance, by Damien Lamberton and Bernard Lapeyre (Chapman and Hall, 1996).

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The mathematics in Steele's book is below the level of, say, Ioannis Karatzas and Steven Shreve (Brownian Motion and Stochastic Calculus, 2 nd ed., Springer-Verlag, 1991), but above that of Bjork. What it lacks in fullblown mathematical rigor and detail, it makes up for in developing one's intuition for the results. (To me, the latter is more valuable.) In fact, hands down, it is the best book I have read for guiding one's intuition in any mathematical topic, not just stochastic calculus. Plus, Steele encourages the reader to use George Polya's "looking back" technique, to guess the answer before proving it, and to further one's understanding through end-ofchapter exercises.

The book has one failing: There are numerous typos-many in the mathematics itself. For example, the definition of the probability density function of the multivariate normal is incorrect, as is a second proof of the Brownian motion characterization theorem. Despite the typos, I still recommend the book for a fine coverage of stochastic calculus.

Virginia R. Young

Associate Professor

University of Wisconsin

School of Business

975 University Ave.

Madison, WI 53706

e-mail: vyoung@bus.wisc.edu

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Stochastic Calculus and Financial Applications

North American Actuarial Journal, Apr 2002 by Young, Virginia R

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