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The study of both sensitivity analysis and differentiability of thestochastic flow of a reflected process in a convex polyhedral domain ischallenging because the dynamics are discontinuous at the boundary of thedomain and the boundary of the domain is not smooth. These difficulties can beaddressed by studying directional derivatives of an associated extendedSkorokhod map, which is a deterministic mapping that takes an unconstrainedpath to a suitably reflected version. In this work we develop an axiomaticframework for the analysis of directional derivatives of a large class ofLipschitz continuous extended Skorokhod maps in convex polyhedral domains withoblique directions of reflection. We establish existence of directionalderivatives at a path whose reflected version satisfies a certain boundaryjitter property, and also show that the right-continuous regularization of sucha directional derivative can be characterized as the unique solution to aSkorokhod-type problem, where both the domain and directions of reflection vary(discontinuously) with time. A key ingredient in the proof is establishingcertain contraction properties for a family of (oblique) derivative projectionoperators. As an application, we establish pathwise differentiability ofreflected Brownian motion in the nonnegative quadrant with respect to theinitial condition, drift vector, dispersion matrix and directions ofreflection. The results of this paper are also used in subsequent work toestablish pathwise differentiability of a much larger class of reflecteddiffusions in convex polyhedral domains.

On directional derivatives of Skorokhod maps in convex polyhedral domains