Synopses & Reviews
This book gives an introduction to the basic theory of stochastic calculus and its applications. Examples are given throughout the text, in order to motivate and illustrate the theory and show its importance for many applications in e.g. economics, biology and physics. The basic idea of the presentation is to start from some basic results (without proofs) of the easier cases and develop the theory from there, and to concentrate on the proofs of the easier case (which nevertheless are often sufficiently general for many purposes) in order to be able to reach quickly the parts of the theory which is most important for the applications. For the 6th edition the author has added further exercises and, for the first time, solutions to many of the exercises are provided. Apart from several minor corrections and improvements, based on useful comments from readers and experts, the most important change in the corrected 5th printing of the 6th edition is in Theorem 10.1.9, where the proof of part b has been corrected and rewritten. The corrected 5th printing of the 6th edition is forthcoming and expected in September 2010.
Review
Aus den Rezensionen zur 6. Auflage: "Dieser Klassiker ... ist sowohl als Lehrbuch als auch zum Selbststudium hervorragend geeignet. Ausgehend von sechs Problemstellungen ... wird in gut verständlicher Weise ... die Theorie stochastischer Differentialgleichungen entwickelt. ... Am Ende eines jeden Kapitels sind zahlreiche Übungsbeispiele zu finden. Die vorliegende 6. Auflage unterscheidet sich von der vorigen vor allem durch zusätzliche Übungsbeispiele sowie Lösungshinweise zu einer Auswahl davon. ... Mittlerweile ist das Buch auch in elektronischer Form als e-book beim Verlag erhältlich." (H. Albrecher, in: IMN - Internationale Mathematische Nachrichten, 2006, Issue 203, S. 36)
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)
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)
Synopsis
This book gives an introduction to the basic theory of stochastic calculus and its applications. Examples are given throughout the text, in order to motivate and illustrate the theory and show its importance for many applications in e.g. economics, biology and physics. The basic idea of the presentation is to start from some basic results (without proofs) of the easier cases and develop the theory from there, and to concentrate on the proofs of the easier case (which nevertheless are often sufficiently general for many purposes) in order to be able to reach quickly the parts of the theory which is most important for the applications. For the 6th edition the author has added further exercises and, for the first time, solutions to many of the exercises are provided. Apart from several minor corrections and improvements, based on useful comments from readers and experts, the most important change in the corrected 5th printing of the 6th edition is in Theorem 10.1.9, where the proof of part b has been corrected and rewritten.
Synopsis
This book gives an introduction to the basic theory of stochastic calculus and its applications. Examples are given throughout the text, in order to motivate and illustrate the theory and show its importance for many applications in e.g. economics, biology and physics. The basic idea of the presentation is to start from some basic results (without proofs) of the easier cases and develop the theory from there, and to concentrate on the proofs of the easier case (which nevertheless are often sufficiently general for many purposes) in order to be able to reach quickly the parts of the theory which is most important for the applications. For the 6th edition the author has added further exercises and, for the first time, solutions to many of the exercises are provided.
Table of Contents
Introduction.- Some Mathematical Preliminaries.- Itô Integrals.- Itô Formula and the Martingale Representation Theorem.- Stochastic Differential Equations.- The Filtering Problem.- Diffusions: Basic Properties.- Other Topics in Diffusion Theory.- Applications to Boundary Value Problems.- Applications to Optimal Stopping.- Application to Stochastic Control.- Application to Mathematical Finance.- Appendix A: Normal Random Variables.- Appendix B: Conditional Expectations.- Appendix C: Uniform Integrability and Martingale Convergence.- Appendix D: An Approximation Result.- Solutions and Additional Hints to Some of the Exercises.- References.- List of Frequently Used Notation and Symbols.- Index.