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.
