  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.
