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Stochastic Differential Equations: An Introduction with Applications (Universitext) Paperback – September 22, 2010

4 out of 5 stars 24 customer reviews

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Editorial Reviews

Review

From the reviews of the fifth edition:

"This is a highly readable and refreshingly rigorous introduction to stochastic calculus. … This is not a watered-down treatment. It is a serious introduction that starts with fundamental measure-theoretic concepts and ends, coincidentally, with the Black-Scholes formula as one of several examples of applications. This is the best single resource for learning the stochastic calculus … ." (riskbook.com, 2002)

From the reviews of the sixth edition:

"The book … has evolved from a 200-page typewritten booklet to a modern classic. Part of its charm and success is the fact that the author does not bother too much with the (for the novice) cumbersome rigorous theory … . This does not mean that the book is not rigorous, it is just the timing and dosage of mathematical rigour … that is palatable for undergraduates … . a highly readable account, suitable for self-study and for use in the classroom." (René L. Schilling, The Mathematical Gazette, March, 2005)

"This is the sixth edition of the classical and excellent book on stochastic differential equations. The main difference with the next to last edition is the addition of detailed solutions of selected exercises … . This is certainly an excellent idea in view to test its ability of applications of the concepts … . certainly one of the best books on the subject, it will be very helpful to any graduate students and also very valuable for any analysts of financial market." (Stéphane Métens, Physicalia, Vol. 26 (1), 2004)

"This is now the sixth edition of the excellent book on stochastic differential equations and related topics. … the presentation is successfully balanced between being easily accessible for a broad audience and being mathematically rigorous. The book is a first choice for courses at graduate level in applied stochastic differential equations. The inclusion of detailed solutions to many of the exercises in this edition also makes it very useful for self-study." (Evelyn Buckwar, Zentralblatt MATH, Vol. 1025, 2003)

About the Author

<P>Bernt Oksendal is a proven Springer author. His book, Stochastic Differential Equations: An Introduction with Applications, has sold over 10,000 copies and has been translated into Japanese and Chinese.</P>
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Product Details

  • Series: Universitext
  • Paperback: 379 pages
  • Publisher: Springer; 6th edition (March 4, 2014)
  • Language: English
  • ISBN-10: 3540047581
  • ISBN-13: 978-3540047582
  • Product Dimensions: 6.1 x 0.9 x 9.2 inches
  • Shipping Weight: 1.4 pounds (View shipping rates and policies)
  • Average Customer Review: 4.0 out of 5 stars  See all reviews (24 customer reviews)
  • Amazon Best Sellers Rank: #405,407 in Books (See Top 100 in Books)

Customer Reviews

Top Customer Reviews

By J. Lim on May 13, 2005
Format: Paperback
This my recommendation for people who want to learn stochastic calculus for the first time. The virtue of this book is that it keeps matters simple,well grounded, and intuitive enough to hook the newcomers in the subject. Once you get comfortable enough and want to learn technical detail necessary for scholarly research, there are other excellent expositions such as Karatzas and Shreve(1998) and Protter(1990). Some reviews complained that this book is limited to stochastic integration with respect to Brownian motion, but that is precisely why I recommend this book. By starting with Browning motion readers can form concrete mental image of stochastic integration and get ready to stride to more general setting if necessary.

Another virtue of this book is the plenty (easy) exercise problems. Working through them is perhaps the best way to learn stochastic calculus.
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It's actually a very good book if you need to learn the topic quickly, armed with a good background in probability theory you will have no difficulty getting through the first 1/3 of the book and gain a working knowledge of SDEs, Ito calculus etc. IT is at times concise in the sense that it lacks motivation etc., but the exposition is such that this presents no major hurdles, as the proofs are clear and short, there are very few errors, except the ones mentioned by the reviewer below, which I should double-check again because I didn't really use this book for its feynman-kac formula (there are better books out there for that). An excellent feature of the book, for those wanting examples from physics and other applied fields, are the problems at the end of chapters. You should definitely give it a try, many of them present the necessary motivation (solutions are at the end of the book). Despite the criticism below, which I consider minor (i.e. it could easily be fixed in a subsequent edition), it is a standard textbook for SDEs, which many respectable mathematicians recommend. Books should be judged by how many times they are quoted by experts, and this book certainly has been cited many times.
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Format: Paperback
If calculus is to real analysis then this book is an attempt at filling in _____ is to stochastic analysis. Stochastic analysis is a difficult topic and a simplified introduction with minimal prerequisites is a great goal. However, this book has not fullfilled its promise.

There are a number of complaints to be made about this book. Most importantly is that in his attempt at simplification, Oksendal frequently chooses shedding (important) details over properly motivating a new concept. I found this particularly true in his exposition of generators. The book is poorly also organized: a number of topics are arbitrarily split into different chapters, important ideas hide inside of examples, etc.

While this is not my favorite book by any means, there is currently no replacement for it. Jumping directly into a book like Karatzas&Shreeve can be daunting. I would recommend getting a used copy. Also, previous editions seem to be very nearly identical to the current edition.

I also recommend checking out Rogers&Williams "Diffusions, Markov Process, and Martingales" Vols I&II.
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The exposition is correct and concise, but too dense for someone without an extensive mathematical background.

I would much rather recommend Shreve's Stochastic Calculus for Finance II. Though longer, it is much more well-motivated and gives you a more intuitive feel for the concepts as opposed to Oksendal's full-on theoreical treatment.
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Format: Paperback
A well written book in Mathematics

Stochastic Differential Equations is a branch of mathematics. This book is not just for financial derivatives analysis or modeling. Oksendal first introduces the subject by raising a few stochastic problems (population growth; electric charge in RLC circuit; filtering problems, Dirichlet problems; asset management; optimal portfolio and options pricing) in the first chapter. The subsequent chapters develop notions and techniques which are able to solve wide varieties of stochastic problems (not just those mentioned in the first chapters). The arrangement is impressive in particular for readers who have no previous knowledge about the subject. The readers at least know the target for developing the techniques and would not lose the way when manipulating tons of symbols. Hints and answers to selected problems are invaluable to students for self-study.

To achieve a sound background on stochastic equations is extremely important especially in quantitative finance. It is not an easy job however. QF students may consider going through this book before seriously take Shreve's books on Stochastic Calculus for Finance.
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This is a standard work (it is the one I read when I first started looking at this sort of thing) but having taken it off the shelf recently again, I think it is overrated, for several reasons.

First, it is very notation heavy - TeX has seduced Mr. Oksendahl into all sorts of bad habits - I can very easily imagine that the earlier editions (mine is the 5th), which were written with a typewriter, are much more readable.

Second, the proofs are very formal, developed mostly in terms of classical functional analysis (square integrable real functions, geometry of real Hilbert spaces etc.). From the point of view of rigor this is fine, but from the point of view of intuition, not so much, esp. when combined with the heavyweight notation. In fact note that unless you have a decent background in functional analysis, of the sort you are more likely to pick up in a mathematics degree than a finance degree, then you are going to get precisely nowhere with this book.

I don't want to be too negative, and there is lots of good stuff here - just to warn that Oksendahl is not (as one might think) a royal road to the theory of SDEs (depressingly, it may be that Oksendahl is, nevertheless, the best of the bunch out there - it is certainly, all criticism not-withstanding, more accessible than Karatzas and Shreve).
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