- Series: Graduate Texts in Mathematics (Book 113)
- Paperback: 470 pages
- Publisher: Springer; 2nd edition (August 25, 1991)
- Language: English
- ISBN-10: 0387976558
- ISBN-13: 978-0387976556
- Product Dimensions: 6.1 x 1.1 x 9.2 inches
- Shipping Weight: 1.9 pounds (View shipping rates and policies)
- Average Customer Review: 4.1 out of 5 stars See all reviews (11 customer reviews)
- Amazon Best Sellers Rank: #162,412 in Books (See Top 100 in Books)
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Brownian Motion and Stochastic Calculus (Graduate Texts in Mathematics) (Volume 113) 2nd Edition
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"A valuable book for every graduate student studying stochastic process, and for those who are interested in pure and applied probability. The authors have done a good job."―MATHEMATICAL REVIEWS
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Top Customer Reviews
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 and 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.
The authors begin in chapter 1 with the task of defining martingales and filtrations, with the notion of a stochastic process being adapted to a filtration taking on particular importance. They omit the proof that a process is progressively measurable if and only if it is measurable and adapted, because of the difficulty of the proof, but give a reference where the proof can be found. Continuous-time martingales are defined, with (compensated) Poisson processes given as an example. The Doob-Meyer decomposition and square-integrable martingales are discussed, and the chapter if full of exercises, with solutions provided to some of these at the end of the chapter. Brownian motion is formally defined in the next chapter, with its existence proven using Wiener measure on the space of continuous functions on the positive half line. The discussion in this chapter has to rank as one of the best in print, due to the meticulous and precise manner in which the material is presented. The Markov property of Brownian motion is proven, along with a good presentation of the Levi modulus of continuity. Readers working in constructive quantum field theory will see their usual construction of Wiener measure in the second exercise of the chapter. Those working in that area are used to seeing (conditional) Wiener measure defined on a collection of cylinder sets, which is then extended to the Borel subsets . Such a construction is done in this book, but the approach is somewhat different than what physicists normally see in quantum field theory.
The theory of stochastic integration is presented in Chapter 3, and it is superbly written. The authors are careful to distinguish the theory of integration for stochastic processes from the ordinary one with emphasis on the actual computation of stochastic integrals. The reader is first asked to explore the Stratonovitch and Ito integrals in an exercise., and then a thorough treatment is given by the authors later in the chapter. The authors point out the differences between the Ito and Stratonovich integrals, with the latter being defined for a smaller class of functions than the former. The important Ito rule for changing variables is discussed, and then used to give the Kunita-Watanabe martingale characterization of Brownian motion. Physicists involved in constructive quantum field theory will appreciate the discussion of the Trotter existence theorem in this chapter.
The connection of Brownian motion with partial differential equations, so familiar to physicists via the heat equation, is the subject of the next chapter. These equations give the transition probabilities of the stochastic process, and are studied here first in the context of harmonic analysis, namely the classical Dirichlet problem. This is followed by a beautiful treatment of the one-dimensional heat equation and the Feynman-Kac formulas. Those readers working in constructive quantum field theory will see the Green's function lurking in the background.
The very important topic of stochastic differential equations is outlined in chapter 5, with emphasis placed on the study of diffusive processes. The solutions of these equations have an immense literature, and the authors do not of course overview all of it, but do give a useful introduction. Both strong and weak solutions are discussed, with the Girsanov and Yamada-Watanabe techniques used throughout. Explicit solutions are given for linear stochastic differential equations, such as the Ornstein-Uhlenbeck process governing the Brownian motion of a particle with friction. Financial engineers will appreciate the discussion of the applications of this formalism to option pricing and the Merton consumption theory in this chapter. Options pricing is cast in martingale terms, and then the usual Black-Scholes equation is derived from this. The notorious Hamilton-Jacobi-Bellman equation is discussed in the consumption/investment problem, and the authors show how to employ techniques for solving this problem instead of solving this difficult nonlinear equation. The authors give a hint of the important Malliavin calculus in the Appendix and give references for the reader.
The last chapter of the book is more specialized than the rest and deals with the Levy theory of Brownian local time. This theory does have a connection with the theory of jump processes, which are currently very important in financial and network modeling. The authors do a fine job of explaining how Poisson random measures permit the event bookkeeping in these jump processes. Their discussion is applied to the computing of the transition probabilities for a Brownian motion with two-valued drift.