# The Malliavin Calculus and Related Topics

Part of the Probability, its Applications book series (PIA)

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Part of the Probability, its Applications book series (PIA)

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.

Anticipating stochastic calculus Brownian motion Gaussian processes Girsanov theorem Malliavin Calculus Markov processes Markov property Skorohod integral Stochastic differential equations Stochastic partial differential equations fractional Brownian motion

- DOI https://doi.org/10.1007/3-540-28329-3
- Copyright Information Springer-Verlag Berlin/Heidelberg 2006
- Publisher Name Springer, Berlin, Heidelberg
- eBook Packages Mathematics and Statistics
- Print ISBN 978-3-540-28328-7
- Online ISBN 978-3-540-28329-4
- Series Print ISSN 1431-7028
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