Fr\'{e}chet derivatives of expected functionals of solutions to stochastic differential equations

Abstract

In the analysis of stochastic dynamical systems described by stochastic differential equations (SDEs), it is often of interest to analyse the sensitivity of the expected value of a functional of the solution of the SDE with respect to perturbations in the SDE parameters. In this paper, we consider path functionals that depend on the solution of the SDE up to a stopping time. We derive formulas for Fr\'{e}chet derivatives of the expected values of these functionals with respect to bounded perturbations of the drift, using the Cameron-Martin-Girsanov theorem for the change of measure. Using these derivatives, we construct an example to show that the map that sends the change of drift to the corresponding relative entropy is not in general convex. We then analyse the existence and uniqueness of solutions to stochastic optimal control problems defined on possibly random time intervals, as well as gradient-based numerical methods for solving such problems.