MALLIAVIN'S CALCULUS AND STOCHASTIC INTEGRAL REPRESENTATIONS OF FUNCTIONALS OF DIFFUSION PROCESSES.

Reference is made to J. M. C. Clark's formula which incorporates the Frechet differentiable functional. In this paper we extend Clark's formula to the more general class of weakly H-differentiable functionals, and we give a simple proof 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 dy equals m(t,y)dt plus sigma (t,y)db. In doing this, we show that y(t) is weakly H-differentiable if m and sigma have bounded, continuous, first derivatives in y.