Sensitivity Analysis for Reflected Diffusions

Abstract: Differentiability of flows and sensitivity analysis are classical topics in dynamical systems.  However, the analysis of these properties for constrained processes, which arise in a variety of applications, is challenging due to the discontinuous dynamics at the boundary of the domain, and is further complicated when the boundary is non-smooth. We show that the study of both differentiability of flows and sensitivities of constrained processes in convex polyhedral domains can be largely reduced to the study of directional derivatives of an associated map, called the Skorokhod map, and we introduce an axiomatic framework to characterize these directional derivatives. In addition, we establish pathwise differentiability of a large class of reflected Brownian motions in convex polyhedral domains and show that they can be described in terms of certain constrained stochastic differential equations with time-varying domains and directions of reflection.