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**Stochastic calculus and financial applications.**
*(English)*
Zbl 0962.60001

Applications of Mathematics. 45. New York, NY: Springer. ix, 300 p. (2001).

This book gives an introduction to stochastic calculus for Brownian motion with applications in mathematical finance. In fact, the theory of option pricing and hedging in complete Itô process models provides the motivation for a detailed development of the tools required to obtain and understand these results. The major part of the book is about stochastic calculus.

The first 80 pages prepare for this by presenting some main results on discrete- and continuous-time martingales and on Brownian motion. These include the Lévy-Ciesielski construction of Brownian motion via the Haar basis in \(L^2[0,1]\), path properties, the reflection principle and Donsker’s theorem (both without proof) and the Skorokhod embedding. Stochastic calculus then begins in Chapters 6 and 7 with the construction of the stochastic integral with respect to Brownian motion, first in \(L^2\) and then by localization for more general integrands. Itô’s formula and quadratic variation in Chapter 8 are followed by stochastic differential equations in Chapter 9; this contains some solution approaches and the existence and uniqueness theorem for Lipschitz coefficients. Weak solutions are not mentioned.

Chapter 11 gives a detailed discussion of the diffusion equation with three approaches (Fourier transform, series, similarity) to obtain a solution and with uniqueness results by means of maximum principles. Chapter 12 proves the martingale representation theorems of Dudley and Itô and the representation of stochastic integrals as time-changes of Brownian motion. Girsanov’s theorem, including a proof of the Novikov criterion, is in Chapter 13, and the final Chapter 15 presents the Feynman-Kac formula and its applications to option pricing and to obtain the arcsine law for Brownian motion.

Specific topics in mathematical finance are covered in Chapters 10 and 14 and used in addition as examples in several other chapters. Chapter 10 explains the central idea of pricing by absence of arbitrage and gives a detailed derivation and discussion of the Black-Scholes PDE and formula. Chapter 14 treats option pricing and hedging in complete models with Itô process prices, gives another derivation of the Black-Scholes formula via martingale arguments and discusses in detail the choice of trading strategies.

As the preface says, “This is a text with an attitude, and it is designed to reflect, wherever possible and appropriate, a prejudice for the concrete over the abstract.”This is also reflected in the style of writing which is unusually lively for a mathematics book. One drawback of the above attitude is that stochastic integrals are only defined with respect to Brownian motion and this sometimes leads to less general (and less transparent) formulations than might have been possible; one example is Lévy’s characterization of Brownian motion. For my personal taste, the derivations in Chapter 14 were also a bit too much on the computational (as opposed to conceptual) side. But on the whole, the results are presented carefully and thoroughly, and I expect that readers will find that this combination of a careful development of stochastic calculus with many details and examples is very useful and will enable them to apply the whole theory confidently.

The first 80 pages prepare for this by presenting some main results on discrete- and continuous-time martingales and on Brownian motion. These include the Lévy-Ciesielski construction of Brownian motion via the Haar basis in \(L^2[0,1]\), path properties, the reflection principle and Donsker’s theorem (both without proof) and the Skorokhod embedding. Stochastic calculus then begins in Chapters 6 and 7 with the construction of the stochastic integral with respect to Brownian motion, first in \(L^2\) and then by localization for more general integrands. Itô’s formula and quadratic variation in Chapter 8 are followed by stochastic differential equations in Chapter 9; this contains some solution approaches and the existence and uniqueness theorem for Lipschitz coefficients. Weak solutions are not mentioned.

Chapter 11 gives a detailed discussion of the diffusion equation with three approaches (Fourier transform, series, similarity) to obtain a solution and with uniqueness results by means of maximum principles. Chapter 12 proves the martingale representation theorems of Dudley and Itô and the representation of stochastic integrals as time-changes of Brownian motion. Girsanov’s theorem, including a proof of the Novikov criterion, is in Chapter 13, and the final Chapter 15 presents the Feynman-Kac formula and its applications to option pricing and to obtain the arcsine law for Brownian motion.

Specific topics in mathematical finance are covered in Chapters 10 and 14 and used in addition as examples in several other chapters. Chapter 10 explains the central idea of pricing by absence of arbitrage and gives a detailed derivation and discussion of the Black-Scholes PDE and formula. Chapter 14 treats option pricing and hedging in complete models with Itô process prices, gives another derivation of the Black-Scholes formula via martingale arguments and discusses in detail the choice of trading strategies.

As the preface says, “This is a text with an attitude, and it is designed to reflect, wherever possible and appropriate, a prejudice for the concrete over the abstract.”This is also reflected in the style of writing which is unusually lively for a mathematics book. One drawback of the above attitude is that stochastic integrals are only defined with respect to Brownian motion and this sometimes leads to less general (and less transparent) formulations than might have been possible; one example is Lévy’s characterization of Brownian motion. For my personal taste, the derivations in Chapter 14 were also a bit too much on the computational (as opposed to conceptual) side. But on the whole, the results are presented carefully and thoroughly, and I expect that readers will find that this combination of a careful development of stochastic calculus with many details and examples is very useful and will enable them to apply the whole theory confidently.

Reviewer: Martin Schweizer (Berlin)