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 ). 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.