On directional derivatives of Skorokhod maps in convex polyhedral domains

Abstract

The study of both sensitivity analysis and differentiability of the stochastic flow of a reflected process in a convex polyhedral domain is challenging due to the abrupt change in the nature of the dynamics at the boundary and is further complicated because the boundary is not smooth. These difficulties can be addressed by studying directional derivatives of an associated extended Skorokhod map, which is a deterministic mapping that takes an unconstrained path to a suitably reflected or constrained version. In this work, we develop an axiomatic framework for the analysis of directional derivatives of a large class of Lipschitz continuous extended Skorokhod maps in convex polyhedral domains with oblique directions of reflection. We establish existence of directional derivatives at a path whose reflected version satisfies a certain boundary jitter property, and also show that the right-continuous regularization of such a directional derivative can be characterized as the unique solution to a Skorokhod-type problem, where both the domain and directions of reflection vary (discontinuously) depending on the state of the reflected path. A key step in the analysis is the proof of certain contraction properties for a family of (oblique) derivative projection operators. The results of this paper are used in subsequent work to study differentiability of stochastic flows and sensitivity analysis for a large class of reflected diffusions in convex polyhedral domains.