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Stochastic Differential Equations: An Introduction with Applications (Universitext) Paperback – September 22, 2010
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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)
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Top customer reviews
The book is structured by first introducing 6 problems which are solved using the concepts and theory discussed in the chapters that follow. This is an excellent pedagogical tool, that is used to focus the mind on applications, in order to understand the abstract concepts discussed.
The level of mathematics is moderate in difficulty with some proofs omitted (but with references included) for the sake of not veering away too far from the main concepts (and the need to introduce further preliminaries to understand the proof).
There are also exercises included (with some solutions and hints) that allows the reader to solidify the understanding and applications.
The follow-up text is commonly the Karatzas and Shreve book,though its level of difficulty is substantially higher than this text.
The text generally starts with an intuitive example for the chapter and then starts methodically working through the underlying mathematics to get to the meaty results. The exercises are worth the effort as they reinforce the chapter work and offer additional insights.
Today the books about the stochastic equations have superated the interest of the traditional analysis.
The author explicates with competence the definition of the martingale, filter or Markov chain. The applications are about the finance, the control theory, the problem of stopping.