On lipschitz continuity of the solution mapping to the skorokhod problem, with applications

Abstract

The solution m the Skorokhoci Problem defines a deieiminisiic mapping of paths that has been found to be useful in several areas of application. Typical uses of the mapping are construction and analysis of deterministic and stochastic processes that are constrained to remain in a given fixed set, such as stochastic differential equations with reflection and stochastic approximation schemes for problems with constraints In this paper we focus on the case where the set is a convex polyhedron and where the directions along which the constraint mechanism is applied arc possibly oblique and multivalued at corner points. Our goal is to characterize as completely as possible those situations in which the solution mapping is Lipschitz continuous. Our approach is geometric in nature, and shows that the Lipschitz continuity holds when a certain convex set, defined in terms of the normal directions to the faces of the polyhedron and the directions of the constraint mechanism, can be shown to exist. All previous instances known to the authors in which the Lipschitz continuity holds are included as special cases. Thus the paper gives a unified treatment of the problem.