On directional derivatives of Skorokhod maps in convex polyhedral domains

TL;DR

This paper develops an axiomatic framework to analyze directional derivatives of Skorokhod maps in convex polyhedral domains, addressing challenges due to boundary discontinuities and non-smoothness, with applications to reflected Brownian motion.

Contribution

It introduces a new axiomatic approach for directional derivatives of Lipschitz continuous Skorokhod maps in convex polyhedral domains, including boundary jitter conditions and time-varying reflections.

Findings

Existence of directional derivatives under boundary jitter conditions

Characterization of derivatives as solutions to Skorokhod-type problems

Pathwise differentiability of reflected Brownian motion in the nonnegative quadrant

Abstract

The study of both sensitivity analysis and differentiability of the stochastic flow of a reflected process in a convex polyhedral domain is challenging because the dynamics are discontinuous at the boundary of the domain and the boundary of the domain 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 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) with time. A key ingredient in the proof is establishing certain contraction properties for a family of (oblique) derivative projection operators. As an application, we establish pathwise differentiability of reflected Brownian motion in the nonnegative quadrant with respect to the initial condition, drift vector, dispersion matrix and directions of reflection. The results of this paper are also used in subsequent work to establish pathwise differentiability of a much larger class of reflected diffusions in convex polyhedral domains.