Convex duality and the Skorokhod Problem. I
- Cite this article as:
- Dupuis, P. & Ramanan, K. Probab Theory Relat Fields (1999) 115: 153. doi:10.1007/s004400050269
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The solution to the Skorokhod Problem defines a deterministic mapping, referred to as the Skorokhod Map, that takes unconstrained paths to paths that are confined to live within a given domain G⊂ℝn. Given a set of allowed constraint directions for each point of ∂G and a path ψ, the solution to the Skorokhod Problem defines the constrained version φ of ψ, where the constraining force acts along one of the given boundary directions using the “least effort” required to keep φ in G. The Skorokhod Map is one of the main tools used in the analysis and construction of constrained deterministic and stochastic processes. When the Skorokhod Map is sufficiently regular, and in particular when it is Lipschitz continuous on path space, the study of many problems involving these constrained processes is greatly simplified.
We focus on the case where the domain G is a convex polyhedron, with a constant and possibly oblique constraint direction specified on each face of G, and with a corresponding cone of constraint directions at the intersection of faces. The main results to date for problems of this type were obtained by Harrison and Reiman  using contraction mapping techniques. In this paper we discuss why such techniques are limited to a class of Skorokhod Problems that is a slight generalization of the class originally considered in . We then consider an alternative approach to proving regularity of the Skorokhod Map developed in . In this approach, Lipschitz continuity of the map is proved by showing the existence of a convex set that satisfies a set of conditions defined in terms of the data of the Skorokhod Problem. We first show how the geometric condition of  can be reformulated using convex duality. The reformulated condition is much easier to verify and, moreover, allows one to develop a general qualitative theory of the Skorokhod Map. An additional contribution of the paper is a new set of methods for the construction of solutions to the Skorokhod Problem.
These methods are applied in the second part of this paper  to particular classes of Skorokhod Problems.