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Stochastic Calculus and Applications

Samuel N. Cohen and Robert J. Elliott
Publication Date: 
Number of Pages: 
Probability and Its Applications
[Reviewed by
Richard Durrett
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Stochastic calculus is such a broad subject that it is hard to describe. One of the basic objects of study is the stochastic differential equation (SDE), which in its simplest form can be written as \[dX_t=b(X_t)\,dt+\sigma(X_t)\,dB_t\] where \(B_t\) is a one-dimensional Brownian motion. We think of this as the differential equation \(dX_t/dt =b(X_t)\) perturbed by noise (which may or may not be small). These processes arise as large population limits in population genetics and epidemiology, but their most famous use is in mathematical finance to model stock prices. Here the authors concentrate on applications to optimal control and filtering, but there are 19 chapters to be covered before one reaches the applications.

This is the second edition of a book first written in 1982 when the general theory of stochastic integration was in its infancy. The beginnings of that subject can be traced to Paul André Meyer’s 1976 article in Séminare des Probabilités de Strasbourg, an annual series of books published in Springer’s Lecture Notes in Mathematics series in the 1960s and 1970s. Much has happened in the last 30+ years, so the book has grown from 300 to 625 pages. To quote the authors “the new edition of this text takes readers who have been exposed to only basic courses in analysis through the modern general theory of random processes and stochastic integration.”

The journey begins with a two chapter introduction to measure theoretic probability. Chapters 4 and 5 discuss martingales in discrete and continuous time, essential prerequisites for building the theory. A review of the first edition quoted on the book’s cover says that the book can be recommended for graduate students, but I think many of them will have difficulties with chapters 3, 6, and 7, which discuss the measure theoretic difficulties that arise in properly formulating the stochastic integral.

To explain the issues at hand, suppose we are given a martingale \(M(s)\), which we are thinking of as a stock price, and a process \(H(s)\) corresponding to the number of shares we hold at time \(s\). Then the stochastic integral \[X_t –X_0 = \int_0^t H(s)\,dM_s\] gives our profit from time \(0\) to time \(t\). Of course, for mathematical and legal reasons, \(H(s)\) cannot be allowed to look at the future beyond time \(s\). The various versions of the notion that \(H(s)\) can be determined from what is known at time \(s\) (optional and predictable process) are the subject of chapter 7. This topic is somewhat delicate because there are uncountably many times, so if there is a null set for each one the union can have positive probability.

Chapters 8–14 develop stochastic integration. Since ordinary integration is a special case it begins with a discussion of integration with respect to processes of finite variation, i.e., signed measures. The next four chapters give the heart of the theory: semimartingales and the quadratic variation are defined. The Doob-Meyer decomposition, Kunita-Watanable inequality, Burkholder-David-Gundy inequality and Itô’s formula are proved. This listing may not mean much to people who don’t know the subject but at least you can see that its development required the efforts of a number of famous probabilists.

Chapters 15–18 develop the theory of stochastic differential equations. When the coefficients \(b\) and \(\sigma\) are Lipschitz then the SDE at the beginning of this review can be solved by Picard iteration. This approach dates back to Itô in the 1940s. Starting from this result one can expand the set of equations that can be solved by using Girsanov’s formula to change the drift \(b\) and by considering weak solutions. Of course one is also concerned with uniqueness of solutions. When that holds the process has the Markov property: given the current state the rest of the path is irrelevant for predicting the future.

Chapter 19 is devoted to the topic of Backward Stochastic Differential Equations. In these equations the terminal value is given and one works backwards. The difficulty in that is that one is looking for a solution so that the value at time \(t\) is determined by the value of the process on \([0,t]\). That’s about all I know… despite hearing several talks on the topic. The importance of this idea can be seen from the fact that the 1990 paper of Pardoux and Peng has been cited 2118 times; the difficulty/obscurity of this topic from the fact it has no Wikipedia page.

As the reader can guess from my summary, this is not a book for undergraduates. Indeed, I doubt if a first year graduate student can master the material. If they do then they will likely be a third year graduate student when they finish. There are many books on stochastic calculus. Many try to be accessible to Master’s students in finance, so they sweep many details under the rug. Others, such as the ones by Karatzas and Shreve, and Oksendal, give reasonably elementary treatments of the case of continuous integrators. However, few books tackle the case with jumps as this one does, so even if you already have a shelf full of books on stochastic calculus this one could be a valuable addition to your library.

Richard Durrett taught at UCLA and Cornell before he came to Duke in 2010. He is a member of the National Academy of Science, who for the last thirty years has used probability to study problems that arise from ecology, genetics, and cancer modeling.

See the table of contents in the publisher's webpage.