The Wiener space derivative for functionals of diffusions on manifolds

Abstract

The Clark-Haussmann formula, giving the integrand in the stochastic integral representation of a functional L(H( omega )), where H=(xt)t epsilon (0,T) is a diffusion process in Rn and L is a mapping from C((0, T),Rn) to R, is equivalent to a computation of the Wiener space derivative of the functional omega to L(H( omega )). It involves the Frechet derivative of L. The author is concerned with the case of diffusions on a manifold M and with a functional L:C((0, T),M) to R; now the Frechet derivative is no longer available. They show that a similar formula can be obtained involving a family of scalar measures nu x and 1-forms qx(t) associated with L. The proof uses stochastic flow theory and the manifold structure of the space C((0, T),M).