Brownian Motion and Stochastic Calculus (Graduate Texts in Mathematics (113)) 2nd Edition

by Ioannis Karatzas (Author), Steven Shreve (Author)

22 ratings

ISBN-13: 978-0387976556

ISBN-10: 0387976558

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A graduate-course text, written for readers familiar with measure-theoretic probability and discrete-time processes, wishing to explore stochastic processes in continuous time. The vehicle chosen for this exposition is Brownian motion, which is presented as the canonical example of both a martingale and a Markov process with continuous paths. In this context, the theory of stochastic integration and stochastic calculus is developed, illustrated by results concerning representations of martingales and change of measure on Wiener space, which in turn permit a presentation of recent advances in financial economics. The book contains a detailed discussion of weak and strong solutions of stochastic differential equations and a study of local time for semimartingales, with special emphasis on the theory of Brownian local time. The whole is backed by a large number of problems and exercises.

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Editorial Reviews

Review

Second Edition

I. Karatzas and S.E. Shreve

Brownian Motion and Stochastic Calculus

"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

Product details

Publisher : Springer; 2nd edition (August 25, 1991)
Language : English
Paperback : 493 pages
ISBN-10 : 0387976558
ISBN-13 : 978-0387976556
Item Weight : 3.35 pounds
Dimensions : 6.1 x 1.12 x 9.25 inches

Best Sellers Rank: #767,035 in Books (See Top 100 in Books)
- #340 in Physics of Mechanics
- #355 in Mechanics
- #411 in Calculus (Books)

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22 ratings

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3 star		22%
2 star		6%
1 star 0% (0%)		0%

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Not an introduction- a very difficult text

Reviewed in the United States on January 14, 2012

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This book is an excellent text on stochastic calculus. As is commonly done, the text focuses on integration with respect to a Brownian motion.

However, there are several important pre-requisites: the reader must be intimately familiar with measure theory, probability theory and stochastic processes. For those new to stochastic calculus it is generally recommended to read Oksendal's book on stochastic differential equations and then come back to Karatzas and Shreve.

Please be warned that even with reasonable background to probability theory and stochastic calculus, this text is very difficult to understand mathematically- it requires a certain level of dedication from the reader if the book is to be read back to back rather than act as a reference

4 people found this helpful

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Paul Thurston

Massive Exercise to the Reader

Reviewed in the United States on March 30, 2005

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This book isn't really the place to start learning about stochastic calculus. Get Oskendal's Stochastic Differential Equations: An Introduction with Applications for this.

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.

135 people found this helpful

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Justin

Caution: Print-on-demand copy

Reviewed in the United States on October 15, 2016

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This is the "Springer Study Edition", apparently a one-time imprint for cheap print-on-demand copies of otherwise out-of-print books. Had I known this (and it does not seem to be mentioned anywhere in the item description), I probably would have sprung for a used copy of the hardcover at a similar price.

As a study copy it should suffice -- the inner margins are a bit tight, but nowhere near as bad as some PoD knockoffs. It just would have been nice to know what I was paying for.

5 people found this helpful

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Chao Yang

Five Stars

Reviewed in the United States on December 29, 2014

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Nice texts for graduate students.

One person found this helpful

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Juan Diluca Haltrich

ok but....

Reviewed in the United States on March 28, 2013

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the book met my expectations, but the rating comes from the condition in which i received the book, which is poor to say it nicely.
2 corners of the book were damaged, and the only reasons i didn't try to give it back were because: 1. it's not that big of a problem and 2. it wasnt severely damaged.
but i will seriously consider these types of problems next time i order another book.

One person found this helpful

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Applied Math Student

A Must

Reviewed in the United States on January 14, 2005

If you want to learn about stochastic calculus, this is the gold standard. Certainly a challenge, but if you can answer all the questions posed in the book you will have a very thorough knowledge of BM, stochastic integration with respect to BM, SDE's, and the SDE/PDE relationship.

While this is a great book, I do have a couple complaints. First of all there a points in the book where overly complicated notation can obscure the point being made. Secondly, since the book does not cover semimartingale integration, topics like the quadratic variation process (which are still important for things like representing a martingale as a time-changed BM) are spread throughout the book and can be difficult to find.

I recommend reading Rogers&Williams "Diffusions, Makov Processes, and Martingales" Vols I&II in addition to this book. Not only does it give a second (and sometimes easier) point of view on the somewhat difficult topic of stochastic analysis, but also covers some topics in greater generality including integration wrt a general semimartingale.

Finally, a warning to those who have interest but are not proper mathematics students. This book presupposes a fair amount of mathematical maturity - if you don't have a good understanding of real analysis and measure-theoretic probability you probably won't understand anything.

18 people found this helpful

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Top reviews from other countries

George C

comprehensive

Reviewed in the United Kingdom on September 8, 2020

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Absolute classics. Very comprehensive and rigorous treatment of the subject.

Amazon Customer

The content of this book is perfect. However

Reviewed in the United Kingdom on November 15, 2015

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The content of this book is perfect . However, the quality of this book is not that good, when I compared the book I purchased with the same book I borrowed from my university's library. It is easily to see that the printing on the cover is little bit blurred. I am not sure this issue is due to the different version of printing or I was 'lucky" to buy an illegal-copy. I need your reply and meanwhile I need to go somewhere to double check my intuition is right or wrong.

Amazon Customer

One of the best books ever written on stochastic calculus

Reviewed in the United Kingdom on December 30, 2016

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One of the best books ever written on stochastic calculus. Easy to read, many exercises with some contained solutions to engage the reader and the book has a clear structure, ending with Paul Levy's theory of local time.

Shilling

95% new

Reviewed in the United Kingdom on October 2, 2013

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Can't believe that I got this nearly brand new book with such a low price. It's really worth the money.

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Workbook

Reviewed in Germany on April 3, 2014

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I do not like that the authors state so many involved facts as problems and exercises. Since they later use many of these results in the main text, they should work them out themselves. Sometimes hints might be given, but since the topic is technically and conceptually difficult, it will often not be possible for the reader to write it down correctly.
Problems are of paramount importance in mathematics and that is why the student deserves handpicked exercises. In this case I have the feeling that whenever the authors do not want to elaborate on cumbersome facts, they state an exercise or a problem. This also makes looking up facts a frustrating experience, since it is cumbersome to check how they proved a certain theorem, only to be referred to some exercises or problems earlier in the book.
The book is best suited for the already advanced reader, who takes this topic very seriously and wants to challenge himself.
Besides the mentioned flaw the authors try hard to communicate the major ideas and also address applications e.g. in mathematical finance.

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