2016/02/04 by David Lipshutz, Lipshutz, David, Kavita Ramanan +1 · 1 citation
Economics, Econometrics and Finance · Mathematics · #60G17 #90B15 (Secondary) #90C31 #93B35 (Primary) #FOS: Mathematics #Mathematical Biology Tumor Growth #Point processes and geometric inequalities #Probability (math.PR) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.1602.01860
openalex publication_date 2016/02/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The study of both sensitivity analysis and differentiability of the\nstochastic flow of a reflected process in a convex polyhedral domain is\nchallenging because the dynamics are discontinuous at the boundary of the\ndomain and the boundary of the domain is not smooth. These difficulties can be\naddressed by studying directional derivatives of an associated extended\nSkorokhod map, which is a deterministic mapping that takes an unconstrained\npath to a suitably reflected version. In this work we develop an axiomatic\nframework for the analysis of directional derivatives of a large class of\nLipschitz continuous extended Skorokhod maps in convex polyhedral domains with\noblique directions of reflection. We establish existence of directional\nderivatives at a path whose reflected version satisfies a certain boundary\njitter property, and also show that the right-continuous regularization of such\na directional derivative can be characterized as the unique solution to a\nSkorokhod-type problem, where both the domain and directions of reflection vary\n(discontinuously) with time. A key ingredient in the proof is establishing\ncertain contraction properties for a family of (oblique) derivative projection\noperators. As an application, we establish pathwise differentiability of\nreflected Brownian motion in the nonnegative quadrant with respect to the\ninitial condition, drift vector, dispersion matrix and directions of\nreflection. The results of this paper are also used in subsequent work to\nestablish pathwise differentiability of a much larger class of reflected\ndiffusions in convex polyhedral domains.\n