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The Malliavin Calculus and Related Topics

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The Malliavin calculus is an infinite-dimensional differential calculus on a Gaussian space, developed to provide a probabilistic ... Weiterlesen
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Beschreibung

The Malliavin calculus is an infinite-dimensional differential calculus on a Gaussian space, developed to provide a probabilistic proof to Hörmander's sum of squares theorem but has found a range of applications in stochastic analysis. This book presents the features of Malliavin calculus and discusses its main applications. This second edition includes recent applications in finance and a chapter devoted to the stochastic calculus with respect to the fractional Brownian motion.



Klappentext

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 Hörmander'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 Hörmander'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.



Zusammenfassung
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.

Inhalt
Analysis on the Wiener space.- Regularity of probability laws.- Anticipating stochastic calculus.- Transformations of the Wiener measure.- Fractional Brownian motion.- Malliavin Calculus in finance.- Malliavin Calculus in finance.

Produktinformationen

Titel: The Malliavin Calculus and Related Topics
Autor:
EAN: 9783540283294
ISBN: 978-3-540-28329-4
Digitaler Kopierschutz: Wasserzeichen
Format: E-Book (pdf)
Herausgeber: Springer
Genre: Grundlagen
Anzahl Seiten: 382
Veröffentlichung: 27.02.2006
Jahr: 2006
Auflage: 2nd ed. 2006
Untertitel: Englisch
Dateigrösse: 3.2 MB
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