Stochastic integral representation of multiplicative operator functionals of a Wiener process

Abstract:

Let M be a multiplicative operator functional of (X, L) where X is a d-dimensional Wiener process and L is a separable Hilbert space. Sufficient conditions are given in order that M be equivalent to a solution of the linear Itô equation \[ M(t) = I + \sum \limits _{j = 1}^d {\int _0^t {M(s){B_j}} } (x(s))d{x_j}(s) + \int _0^t {M(s){B_0}(x(s))ds,} \] where ${B_0}, \ldots ,{B_d}$ are bounded operator functions on ${R^d}$. The conditions require that the equation $T(t)f = E[M(t)f(x(t))]$ define a semigroup on ${L^2}({R^d})$ whose infinitesimal generator has a domain which contains all linear functions of the coordinates $({x_1}, \ldots ,{x_d})$. The proof of this result depends on an a priori representation of the semigroup $T(t)$ in terms of the Wiener semigroup and a first order matrix operator. A second result characterizes solutions of the above Itô equation with ${B_0} = 0$. A sufficient condition that M belong to this class is that $E[M(t)]$ be the identity operator on L and that $M(t)$ be invertible for each $t > 0$. The proof of this result uses the martingale stochastic integral of H. Kunita and S. Watanabe.