Preface

This monograph is compiled from the notes of a series of ten lectures given

at the NSF-CBMS Conference on Malliavin Calculus and Its Applications at Kent

State University, Ohio, August 7th to 12th, 2008.

The Malliavin calculus or stochastic calculus of variations is an infinite-dimen-

sional differential calculus on the Wiener space that has been developed from the

probabilistic proof of H¨ ormander’s hypoellipticity theorem by Paul Malliavin in

1976 (see the reference [25]). Contributions by Stroock, Bismut, Kusuoka, and

Watanabe, among others, have expanded this theory in different directions.

The main application of Malliavin calculus is to establish the regularity of the

probability distribution of functionals of an underlying Gaussian process. In this

way one can prove the existence and smoothness of the density for solutions to

ordinary and partial stochastic differential equations. In addition to this main

application, the Malliavin calculus has proved to be a powerful tool in a variety

of problems in stochastic analysis. For example, the divergence operator can be

interpreted as a generalized stochastic integral, and this has been the starting point

of the development of the anticipating stochastic calculus. In the last years some

new applications of Malliavin calculus in areas such as central limit theorems and

mathematical finance have emerged.

The purpose of these lectures is to introduce the basic results of Malliavin

calculus and its applications. We have chosen the general setting of a Gaussian

family of random variables associated with an arbitrary separable Hilbert space.

Some of the applications are just briefly introduced, and we recommend the reader

to look over additional references for more details.

In the first three chapters we introduce the fundamental operators: the deriva-

tive operator D; its adjoint δ, called the divergence operator; and the generator of

the Ornstein-Uhlenbeck semigroup, denoted by L. Chapter 4 is devoted to proving

the Meyer inequalities and the continuity of the divergence operator in the Sobolev

spaces. The remaining chapters deal with a variety of applications of Malliavin

calculus. First, in Chapter 5, we establish the general criteria for the existence

and smoothness of densities for functionals of a Gaussian process. In Chapter 6 we

discuss properties of the support of the law of a given Gaussian functional that can

be proved using Malliavin calculus. Chapter 7 deals with the proof of H¨ormander’s

hypoellipticity theorem. In Chapter 8 we discuss the use of the divergence as an

anticipating stochastic integral with respect to the Brownian motion. This chapter

also contains an introduction to the stochastic calculus with respect to the fractional

Brownian motion, using techniques of Malliavin calculus. Chapter 9 presents some

recent applications of Malliavin calculus to derive central limit theorems for mul-

tiple stochastic integrals, and Chapter 10 describes some applications of Malliavin

calculus in mathematical finance.

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