The Malliavin Calculus and Related Topics (Probability and Its Applications)

The Malliavin calculus (or stochastic calculus of variations) is an infinite-dimensional differential calculus on the Wiener space. Originally, it was developed to prove a probabilistic proof to Hörmander's "sum of squares" theorem, but more recently it has found application in a variety of stochastic differential equation problems. This monograph presents the main features of the Malliavin calculus and discusses in detail its connection with the anticipating stochastic calculus. The author begins by developing analysis on the Wiener space, and then uses this to analyze the regularity of probability laws and to prove Hörmander's theorem. Subsequent chapters apply the Malliavin calculus to anticipating stochastic differential equations and to studying the Markov property of solutions to stochastic differential equations with boundary conditions.