The Malliavin Calculus and Related Topics (Probability and Its Applications) - Hardcover

Nualart, David

Hardcover

ISBN 10: 3540283285 ISBN 13: 9783540283287

Publisher: Springer, 2005

Synopsis

There have been ten years since the publication of the ?rst edition of this book. Since then, new applications and developments of the Malliavin c- culus have appeared. In preparing this second edition we have taken into account some of these new applications, and in this spirit, the book has two additional chapters that deal with the following two topics: Fractional Brownian motion and Mathematical Finance. The presentation of the Malliavin calculus has been slightly modi?ed at some points, where we have taken advantage of the material from the lecturesgiveninSaintFlourin1995(seereference[248]).Themainchanges and additional material are the following: In Chapter 1, the derivative and divergence operators are introduced in the framework of an isonormal Gaussian process associated with a general 2 Hilbert space H. The case where H is an L -space is trated in detail aft- s,p wards (white noise case). The Sobolev spaces D , with s is an arbitrary real number, are introduced following Watanabe’s work. Chapter2includesageneralestimateforthedensityofaone-dimensional random variable, with application to stochastic integrals. Also, the c- position of tempered distributions with nondegenerate random vectors is discussed following Watanabe’s ideas. This provides an alternative proof of the smoothness of densities for nondegenerate random vectors. Some properties of the support of the law are also presented.

About this title

Review

From the reviews of the second edition:

"Nualart’s book serves both pedagogic and research needs. On the one hand, it is written to teach the subject. ... On the other hand, the applications in the book are sufficiently broad and in depth that the reader who masters them should be prepared for research. ... Furthermore, the unified approach and the careful statement of technical results in the development of the applications make the text a handy reference for researchers. ... The bibliography is extensive and has been updated." (Daniel Ocone, Mathematical Reviews, Issue 2006 j)

Synopsis

The Malliavin calculus (or stochastic calculus of variations) is an infinite-dimensional differential calculus on a Gaussian space. Originally, it was developed to provide a probabilistic proof to Hormander's "sum of squares" theorem, but it has found a wide range of applications in stochastic analysis. This monograph presents the main features of the Malliavin calculus and discusses in detail its main applications. The author begins by developing the analysis on the Wiener space, and then uses this to establish the regularity of probability laws and to prove Hormander's theorem. The regularity of the law of stochastic partial differential equations driven by a space-time white noise is also studied. The subsequent chapters develop the connection of the Malliavin with the anticipating stochastic calculus, studying anticipating stochastic differential equations and the Markov of solutions to stochastic differential equations with boundary conditions. The second edition of this monograph includes recent applications of the Malliavin calculus in finance and a chapter devoted to the stochastic calculus with respect to the fractional Brownian motion.

Edition details

• Publisher: Springer
• Publication date: 2005
• Language: English
• ISBN 10: 3540283285
• ISBN 13: 9783540283287
• Binding: Hardcover
• Edition number: 2
• Number of pages: 396