Product Description

The second volume concentrates on stochastic integrals, stochastic differential equations, excursion theory and the general theory of processes. These subjects are made accessible in the many concrete examples that illustrate techniques of calculation, and in the treatment of all topics from the ground up, starting from simple cases. Many of the examples and proofs are new; some important calculational techniques appear for the first time in this book.

Product Details

• Amazon Sales Rank: #75812 in Books
• Published on: 2000-09-18
• Number of items: 1
• Binding: Paperback
• 794 pages

Review
'I welcome the paperback edition version of this masterfully written text.' Paul Embrechts, JASA

'The monograph as a whole is warmly recommended to post-PhD students of probability and will be welcomed as a good and reliable reference.' EMS

'... will be read with pleasure and advantage by experts in the field and its applications, as well as by those probabilists and others who wish to learn the subject ... an exciting and enjoyable introduction to the rich ideas of the Itô calculus ... there is nothing dry about this book, for its authors have already breathed life into a vibrant subject.' Mathematics Today

Book Info
A systematic treatment of the subject of diffusions, Markov processes, and martingales, concentrating of stochastic integrals, stochastic differential equations, excursion theory and the general theory of processes. Treats the subject from the ground up, making it easier for students to follow along and apply these concepts to many disciplines. Previous edition: c1994. Softcover.

Customer Reviews

Definitive Approach to Brownian Motion and Stochastic Calculus
In this second volume in the series, Rogers & Williams continue their highly accessible and intuitive treatment of modern stochastic analysis. The second edition of their text is a wonderful vehicle to launch the reader into state-of-the-art applications and research.

The main prerequisite for Volume 2,'Ito Calculus', is a careful study of Volume 1,'Foundations'. The reader may want to prepare for the stochastic differential geometry material in Chapter 5. As a good introduction, I recommend DoCarmo's 'Riemannian Geometry' (a definitive treatment can be found in Spivak's five volume set 'A Comprehensive Introduction to Differential Geometry'). Note that this second volume is not entirely self-contained and the reader is given copious references to the research literature to augment the main thread.

The book begins with Chapter 4, which develops the Ito theory for square-integrable semimartingale integrators which are either of bounded variation or are continuous.
The chapter begins with a definition of the allowable integrands. These are the so called 'previsible' processes and this notion generalizes the concept of left-hand continuity. Some authors (Karatzas & Shreve 'Brownian Motion and Stochastic Calculus, 2nd edition, page 131) refer to such integrands as 'predictable'.
As a warm-up into the full theory, the authors present Ito calculus from the Riemann-Stieltjes point-of-view for integrators of bounded variation. Applications to Markov chains are studied which foreshadow the strong Markov process applications derived later on from a more full-fledged theory.

The main simplification that the authors derive from continuity assumption is the implicit agreement of the optional quadratic variation process and the Doob-Meyer predictable quadratic variation process. This helps streamline the presentation of the more full-fledged theory and allows the reader to get the main applications more quickly.

All the key results from the classical Ito theory are presenting in this chapter, including Integration by Parts, Ito's Formula, Levy's characterization Theorem, the martingale transformation Theorem, Girsanov's Theorem and Tanaka's formula for Brownian Local Time. There is also a nice treatment of the Stratonovich calculus and its relation to the Ito theory.

For readers of Volume 1, the material in Volume 2, Chapter 5 is the long awaited development of stochastic differential equation techniques to explicitly construct Markov processes whose transition semigroups satisfy the Feller-Dynkin hypotheses.

After some motivating examples of diffusions from physical systems and control theory (including the ubiquitous Kalman-Bucy filter), the authors focus on strong solutions of SDE's. Ito's existence theorem, which was inspired by a Picard-type algorithm from the theory of classical PDEs, is presented for SDE's with locally lipschitz coefficients. As a really terrific application of Ito's existence theorem, Rogers & Williams introduce the notion of a Euclidean stochastic flow.

Next up, the discussion turns to weak solutions of SDEs, the martingale problem of Stroock and Varadhan. Existence of solutions of the martingale is established with a nice probability measure convergence argument. This treatment really gives the flavor of the Stroock-Varadhan theory and is much more accessible than the full-blown Krylov results found in the Ethier & Kurtz text 'Markov Processes Characterization and Convergence'.

For me, the real highlight of Chapter 5 is the wonderful section introducing stochastic differential geometry. Diffusions on n-dimensional manifolds are introduced and the interplay between Ito and Stratonovich calculus is carefully studied. Examples of diffusions on Riemannian manifolds are studied in some detail.

Chapter 6 extends the Ito theory developed in Chapter 4 to general square-integrable semimartingale integrators. The Doob-Meyer decomposition is explored and the divergence between predictable quadratic variation and optional quadratic variation [M] for a square integrable (local) martingale is studied. Next, [M] is generalized sufficiently to complete the development of the Ito calculus. The general Ito Formula is applied to such problems as the Kalman-Bucy Filter and the Bayesian Filter of Kallianpur-Striebel.

The book wraps up with an introduction to excursion theory. The premise here is that we want to study those times for which a Markov process visits a compact set. The theory leads to some nice results, including a proof of the embedding theorems of Skorokhod and Azema-Yor along with applications to potential theory and the general study of local time.

A Great Book
This book and its companion volume are a well organized and relatively easy-to-read introduction to a wide variety of ideas in stochastic processes. It is not only a great reference (I always keep it on my desk) but it also has a solid expositional style that fully motivates concepts as they are introduced. The Ito Calculus volume goes deeper than a number of other books on topic including information on integration wrt to a general semimartingale instead of just BM and even an introduction to stochastic calculus on manifolds. My only complaints about the book are that it is separated into two volumes which can be kind of a pain and that its coverage of the SDE/PDE relationship is weak. I would recommend reading Karatzas&Shreeve in addition to this book to fill in some the SDE/PDE details and to get another point of view on the somewhat difficult topic of stochastic analysis

Pretty accessible
The parts of this book I've read have been clear and accessible for someone with an undergraduate degree in mathematics and some knowledge of stochastic processes. It doesn't needlessly multiply the jargon like some books, and it focuses mainly on the one-dimensional case so that the intuition isn't constantly obscured by matrix notation. Many subjects also have chatty introductions that offer intuition and a bit of relief from the hard work involved in learning this subject.