Malliavin's calculus and stochastic integral representations of functional of diffusion processes^†

Abstract

If F is a Frechet differentiable functional on is a Brownian motion, and clark's formula states that where is the measure defining the Frechet derivative of F at b.In this paper we extend Clark's formula to the more general class of weakly H-differentiablefunctionals, and we give a simple proff based on Malliavin's calculus. again using Malliavin calculus techniques, we also derive Haussmann's stochastic integral representation of a function F(y) of the diffusion process In doing this, we show that is weakly H-differentiable if m and have bounded, continuous, first derivatives in y.