Continuous Stochastic Calculus with Applications to Finance. (Book Reviews)
Rheinlaender, Thorsten, Journal of the American Statistical Association
Michael MEYER, Boca Raton, FL: Chapman and Hall, 2001. ISBN 1-58488-234-4. vi+319 pp. $89.95.
The scope of the book is first to develop rigorously the theory of continuous-time martingales and stochastic integration, and then, in the final third, to give some applications of these methods to mathematical finance. By collecting this material in one textbook, the author claims to fill a gap in the existing literature. However, there actually are a couple of excellent texts on this subject [e.g., Elliott and Kopp (1999); Karatzas and Shreve (1998); Lamberton and Lapeyre (1996); Steele (2001), to name but a few].
The prerequisites for reading this text are minimal. Only a basic knowledge of measure theoretic probability and Hilbert space theory is assumed. The book is very readable. Proofs are carefully written down and given in every detail, although exercises are not provided.
The treatment starts with a development of martingale theory. Brownian motion is presented as fundamental example of a continuous martingale, followed by an introduction to stochastic integration with respect to continuous martingales. Topics of stochastic analysis relevant to finance as the change of measure technique and martingale representation theorems are presented in detail. The material of the first three chapters is fairly standard and can be found in many textbooks. I am not that happy with Meyer's treatment of integration with respect to vector-valued continuous semimartingales, however. This is introduced only as sum of componentwise integrals, a concept of only limited use. Even in a Brownian setting, componentwise stochastic integration is not the right concept in many cases. [Compare example III.4.10 of Jacod and Shiryaev (1987) or the discussion in Chatelain and Stricker (1994), where the authors stress the relevance of this point for finance.]
In the final chapter, the author turns to applications to finance. After the obligatory Black-Scholes price for an European call option is derived, the general market model is introduced. This is the well-known model where the noise source is a d-dimensional Brownian motion. Basic concepts such as arbitrage or change of numeraire are carefully introduced. However, I miss a discussion of the very basic notion of market completeness. …