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Elements of Stochastic Calculus and Analysis

Daniel W. Stroock
Publication Date: 
Number of Pages: 
[Reviewed by
Russel J. Hendel
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The Elements presents the theory of stochastic calculus and analysis using a historical approach, focusing on the core ideas which motivated and developed the material. The author acknowledges that as the theory developed, new ideas were introduced (some of which, in fact, are also covered in the book) providing greater clarity and depth. As the author explains,
For those who are content to master and apply the resulting theory, the origins of the theory are of little importance; for those who are hoping to solve new problems, the initial insight can be of greater value than the finished product.
Consequently, the book is suitable for post-doctorates, researchers, or as a text for a 3 rd -year graduate course.  The author provides several sources for alternate traditional treatments including applications: McKean’s Stochastic Integrals,  Oksendal’s Stochastic Differential Equations: An Introduction with Applications which has gone through half a dozen editions, Ito’s memoir, On Stochastic Differential Equations  and Doob's brief treatment in his Stochastic Processes.
The book has about 30 sections making it suitable for a 1 semester course. The exercises are upper graduate level exercises encouraging research (rather than mastery of techniques). The exercises include such diverse topics as the Ornstein-Uhlenbeck operator, the Levy-Khinchine formula for infinitely divisible laws, the heat equation, Gaussian families, the elliptic strong minimum principle, the mean value property, the Hermite polynomials with their application to stochastic settings, and Girsanov’s theorem.  Despite the author’s criticism (of his own book!), the Elements present several topics that are not usually treated elsewhere including, Wiener’s theory of homogeneous chaos, a novel development of the Stratovich integration theory, applications of the Stratovich theory to derive Wong’s and Zakai’s approximation theory, and applications of the Malliavin calculus to partial differential equations.
While the author disagrees that his own book is likely to go through six editions, every book, consistent with its goals and directions, can typically benefit from future editions. In this case, the book lacks significant mention of applications. Just as the historical origins of a subject can be important in approaching new problems, so too, each application of a subject naturally suggests results and theorems with intuitive interpretation.Thus the author, in a future edition, could mention applications. As a simple example one of the exercises asks for a proof of the Girsanov theorem. Why not add a problem subpart showing how the Girsanov theorem can naturally be used in the theory of finance to elegantly price exotic options as risk-neutral expectations of their discounted payoffs.
As mentioned earlier, the Elements does not seek to provide the finished full-blown theory.  Consequently, Ito’s formula, which the author (correctly) calls the crown jewel, does not appear until chapter 3. Earlier chapters deal with core ideas which formed the foundation for thinking about Stochastic concepts. Thus Chapter 1 presents Kolomogorov’s equations and discusses linear functionals that satisfy the minimum principle and canonical paths. Chapter 2, based on Ito’s approach presents Brownian motion, the Wiener measure and associated concepts. Although other books may mention these concepts also, the novelty of the Elements is to start from these concepts rather than list them as afterthoughts.
Nowhere does the book mention prerequisites. However, from the book, one may infer that reasonable prerequisites are familiarity with functional analysis, measure theory, integration theory, Banach and Hilbert space theory, and operator theory.


Russell Jay Hendel (RHendel@Towson.Edu) holds a Ph.D. in theoretical mathematics and an Associateship from the Society of Actuaries. He teaches at Towson University. His interests include discrete number theory, applications of technology to education, the theory of pedagogy and instructional design, and biblical exegesis.