Malliavin Calculus

Abstract

The stochastic calculus of variation initiated by P. Malliavin is a kind of infinite dimensional differential analysis on the Wiener space. Since N. Wiener constructed in 1923 a mathematical model for Brownian motion, namely the Wiener measure on the space of continuous functions, many attemps have been made to develop a theory of differential analysis for Wiener functionals. Unfortunately, they were not successful since most usual functionals such as Itô integrals and solutions of Itô stochastic differential equations may be not differentiable in the sense of Fréchet, even not continuous as functionals on the Wiener space. In 1976, by virtue of the quasi-invariance of Wiener measure, P. Malliavin introduced a kind of weak differential calculus for Wiener functionals such that the above mentioned important functionals became smooth under his sense of differentiation and thus opened up a new prospect. Using this kind of calculus, he investigated the smoothness of densities of Wiener functionals, invented a nice probabilistic proof to the celebrated Hörmander’s theorem on hypoellipticity of differential operators and thus received widespread attention from mathematical society. This kind of differentiation was defined by “perturbation” of Brownian paths, hence obtained the name “stochastic calculus of variation” and now popularly known as “Malliavin calculus”.