Massive Exercise to the Reader
Reviewed in the United States 🇺🇸 on March 30, 2005
This book isn't really the place to start learning about stochastic calculus. Get Oskendal's
Stochastic Differential Equations: An Introduction with Applications
Even to the prepared reader, this book is exasperating. It is as if the authors came up with an excellent outline for an advanced treatment of this topic. Then they realized that to do all of the material justice, they'd need to have not one, but two 400 page volumes. Their publisher must have balked at that idea, so their solution was to leave out half the detail, forcing each of our poor readers to re-generate the missing 400 pages of needed detail on his/her own. In the opinion of this reviewer, that is exactly what they have done with this text.
Fortunately for us all, there exists a nice two volume (800 page total pages) treatment of this material. Rogers & Williams
Diffusions, Markov Processes, and Martingales: Volume 1, Foundations
Diffusions, Markov Processes and Martingales: Volume 2, Itô Calculus
provide a thorough, accessible exposition with all the needed rigor, generality and detail.
Karatzas & Shreve's treatment of early foundational material is less than helpful to the student. Consider a pair of key results on martingales early on in the text: the optional sampling theorem and the optional stopping theorem. The authors "prove" the optional sampling theorem by appealing to the discrete time results in Chung's
A Course in Probability Theory
and then applying limiting arguments to bootstrap to the continuous time case. Since all of the real "ideas" are in the discrete time case, it's not clear how much of a service the authors' treatment really is. Worse yet, the optional stopping theorem isn't even called out as a theorem, but instead buried as problem.
It is curious to see which topics inspire the authors to spill ink. For example in Chapter 2, we get not one, but 3, yes three different constructions of Brownian motion: convolved heat kernels, Haar interpolation and random walks/Wiener measure. Of course, only the last construction is used going forward and the first two constructions are not brimming over with detail. This is a curious indulgence in a text that is purposefully being stingy with detail. Our poor reader has to pay the price for this indulgence with an extremely terse treatment of the strong Markov property and reflection principle, the Blumenthal Zero-One Law, and other foundational properties of Brownian motion.
Chapter 3 represents the core of the text and develops all the of "greatest hits" including the Ito Integral, Ito's rule, Levy's characterization of Brownian motion, the martingale representation theorem, the Girsanov Theorem and an introduction of Brownian local time. (Brownian local time is further developed in Chapter 6). The development of the Ito Integral is shamelessly sketchy. All the theorems are correctly stated, but the "proofs" offered aren't detailed enough to explain why all of the stated assumptions are needed. When the reader gets to the development of Ito's rule, he/she finds a rude 3 sentence introduction to semi-martingales, a topic which hadn't been explored and never gets more than a passing mention in the authors' text.
Assuming that you've understood everything going on in the text up to this point, Chapter 4 is quite nice. It gives a very intuitive introduction in the role of the Mean Value Theorem as a hook connecting stochastic integrals with classical PDE's. The section on Harmonic functions and the Dirichlet problem is quite nice. The material on the heat equation requires properties of Brownian motion most easily derived from the convolved heat kernels construction. The chapter winds up with a nice treatment of the Feynman-Kac formulas.
After the PDE's material, the reader might develop a sense of hope that the remainder of the exposition will be readily accessible. This is not the case and with the SDE's in Chapter 5, the authors return to their now too familiar terse style as they study strong and weak solutions to stochastic differential equations. At one point, the authors decide to approach the problem by generalizing from functions to functionals without even so much as defining their notion of a functional.
Really, the only good role for this text is as base material for a do-it-yourself "Moore Method" class on stochastic calculus, like they used to do for general topology at the University of Texas. If you completed a Moore-style class this way and wrote up all of your work, you'd have a very fine text covering
diffusions, Markov processes, and martingales.
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