A guide to the stochastic calculus of variations

Abstract

In his path-breaking articles, [43] and [44], P. Malliavin succeeded in obtaining a probabilistic theory of hypoellipticity for second order elliptic and parabolic partial differential equations. Following his lead, Stroock [64,64,66,67], Kusuoka and Stroock [39], Bismut [6], Ikeda and Watanabe [30], Shigekawa [62], and others have developed Malliavin's ideas into an extensive theory for examining existence and regularity of densities for the probability distributions of Wiener functionals, and this theory is now finding an ever increasing arena of applications. Today, the subject is known informally as the 'Malliavin calculus', and formally as the 'stochastic calculus of variations' (hereafter, SCV). The name 'stochastic calculus of variations' is perhaps misleading at first; it refers not to calculus of variations in the traditional sense of minimizing functionals of paths, but to the variations in path spaee used to define certain derivatives of Wiener functionals. Such derivatives, together with a corresponding differential calculus, play a basic role in the theory.