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An Introduction to Stochastic Differential Equations

Lawrence C. Evans
American Mathematical Society
Publication Date: 
Number of Pages: 
[Reviewed by
Robert Riehemann
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Lawrence Evans, winner of the Steele prize and author of the standard graduate book on Partial Differential Equations, has written an interesting and unusual introduction to stochastic differential equations that he aims at beginning graduate students and advanced undergraduates. This is an updated version of his class notes, taught over the years at the University of Maryland, College Park and the University of California, Berkeley.

It certainly is topical and appealing to a wide audience. As recently as October of 2012, Ian Stewart, the well-known mathematician from the University of Warwick, wrote an article in The Guardian entitled The Mathematical Equation that Caused the Banks to Crash, about the relationship of the Black-Sholes-Merton equation to the economic crisis of 2008. Kyoshi Itô won the Gauss prize in November of that year for his invention of the Itô Calculus, used in the derivation of BSM. The Feynman-Kac formula, relating, in a mathematically rigorous way, the solutions of parabolic partial differential equations to stochastic processes, Norbert Weiner’s creation of a useful measure on \(C[0,1]\) and Einstein’s famous paper on Brownian motion in 1905, are all ideas presented in this little book of 150 pages. This is interesting stuff and, because of Evans’s always clear explanations, it is fun too.

Yet it is rather surprising to find so much advanced material in such a short book and one wonders about the assumed student preparation. Evans is straightforward: “I assume my readers to be fairly conversant with measure-theoretic mathematical analysis… I downplay measure theoretic issues … I “prove” many formulas by confirming them in easy cases …” This is “a survey.” This immediately raises the existence question for appropriate students and in turn the philosophical meaning of the advertising blub on the back cover, “… accessible to non-specialists …” My personal experience is that there are two common uses for the idea of “advanced student”:(1) one who has enough talent to succeed and do well in graduate school and (2) one who has already successfully completed graduate level work as an undergraduate. The former is typically used at four year colleges and the latter at research institutions. It is the second sense that is appropriate here. Not only is the construction of Lebesgue measure assumed and the Fatou, Dominated Convergence theorems, and Jensen’s Inequality freely used, but the Stone-Weierstrass density and results from complex analysis are used without explicit reference so that a thorough knowledge of all of baby Rudin or a graduate level analysis text is needed. Results on the smoothness of solutions to parabolic PDE are not developed, of course. It would not hurt to have more than a cursory knowledge of physics in the section on Brownian motion either.

The book is divided into twenty-eight sections, and is clearly intended to be entirely covered in a semester. So approximately one section would be covered per class period in a course that meets twice a week in a standard 14 or 15 week semester. The appropriately titled, “Crash Course in Probability Theory”, would then be covered in seven days. It covers more material (through conditional expectation — using measure theory — and including basic information about Martingales) than I manage in a half-semester using Miller and Miller’s standard engineering probability text. If one compares Evans’s development to that of Friedman’s complete development in Stochastic Differential Equations and Applications, Volume 1, it is clear that the condensation factor is greater than two. The 51 problems at the end of the text are similarly succinct, sometimes only a problem or two for an important idea. There is not much room for examples and several have hints. Because of the rate at which new information is coming to the student, these will be perceived as very difficult. This is not for the faint-hearted by any means.

So what saves the student from drowning in this flood of abstract, unfamiliar material? It is Evans’s clarity, beautiful writing and thoughtful insight. This is present in his PDE text and it is present here. It was present in the classroom when I studied geometric measure theory with him and it is his hallmark. The presentation is clearly and logically organized. Lacunae are appropriately noted. Remarks are also separated off and not buried in the middle of paragraphs so that a student can profitably review material and find an insightful remark when needed. For the “non-expert” there are summaries on “How to remember Itô’s Chain Rule” and other computational tools. The book proceeds by developing probability theory, defining Brownian motion (the Wiener Process) and “white noise”, then developing stochastic integrals and stochastic differential equations. Finally, the last 4 sections are devoted to Feynman-Kac, Optimal Stopping, Options Pricing and the Stratonovich Integral. These are the goals and the applications. What especially interested me was the inclusion of the Stratonovich integral and this relates, in some ways, to issues raised by Stewart in his comments about the 2008 economic crisis.

Evans considers a modeling issue of great practical importance. Precisely, if \(\{\xi^k( \cdot) \}_{k=1}^\infty\) is a sequence of stochastic processes with:

  • \(E(\xi^k (t)) = 0 \ ,\)
  • \(E( \xi^k(t)\xi^k(s) ) := d^k(t-s) \text{ for } d^k(\cdot) \text{ convergent to the Dirac point mass at the origin,} \)
  • \(\xi^k(t) \text{ are gaussian and } \)
  • \(t \mapsto \xi^k(t) \text{ are smooth (and not merely continuous),} \)

then we may consider the problem \[ \left\{ \begin{array}{rl} \dot{X^k} &= d(t) X^k + f(t) X^k \xi^k \\ X^k(0) &= X_0 \end{array} \right. \] This is to be compared with the traditional problem in which \(\xi\) is thought of as “white noise” and is replaced by Brownian motion, where the interpretation becomes \[ \left\{ \begin{array}{rl} dX &= d(t) X\, dt + f(t) X \, dW \\ X(0) &= X_0 \end{array} \right. \] The former (smooth) problem has solutions \[ X^k(t) = X_0 e^{\int_0^t d(s)\, ds + \int_0^t f(s)\xi^k(s)\, ds } \] that converge to the process \[ \hat{X}(t) = X_0 e^{\int_0^t d(s)\, ds + \int_0^t f(s)\, dW } \ . \] The “white noise” problem is a standard Itô solution, \[ X(t) = X_0 e^{\int_0^t d(s)-\frac{1}{2} f^2(s)\, ds + \int_0^t f(s)\, dW } \ . \] Thus the smooth approximations do not yield the canonical Itô result and it is upon the Itô calculus that the Black-Sholes-Merton equation is based. As Evans notes, this means that the Itô problem is “… unstable with respect to changes in the random term \(\xi(\cdot)\).” It is for this reason that the Stratonovich integral is introduced; it is stable in the sense defined above.

It does not seem to be known exactly where the modeling failed that contributed so strongly to the collapse of the economy in 2008. The relevant investigations have not been made. It is known, as noted by Stewart, that the assumption of constant standard deviation is incorrect in the derivation of Black-Sholes-Merton. As is clear from Evans’s remarks on the stability with respect to the noise term, more subtle modeling issues might also be of importance. It is wonderful for students to be exposed to such implications. The world economy was threatened in 2008 and mathematics was directly involved. Raising questions of moral obligation and scientific responsibility are part of every instructor’s duty, so this is exciting. I love this text for making accessible the mathematics that underlies an important issue of such magnitude.

Is the book really accessible to the non-expert as claimed on the back cover? Perhaps, if non-experts include economists and physicists highly trained in mathematics or the casual Ph.D. in mathematics. Citizens in the USA, of course, are inured to hyperbolic advertisements. But it might be better if the generic non-expert wait for the movie, with a script that will doubtless be written by Woody Allen.

Robert Riehemann teaches at Thomas More College in Crestview Hills, Kentucky. He can be reached at