Smoothness of the Solution Operator of Stochastic Differential Equations with Infinite Dimensional Parameters

Abstract

In this paper it is shown that the solution operator for stochastic differential equations depends smoothly on infinite dimensional parameters appearing in the coefficients of the SDEs, if these coefficients themselves depend smoothly on the parameters. This result is a generalization of smoothness results with respect to finite dimensional parameters. The result is used to show differentiability with respect to the coefficients of SDEs. Possible applications in stochastic optimal control and for approximating solutions of SDEs are given.