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Convex duality and the Skorokhod Problem. II
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  • Published: September 1999

Convex duality and the Skorokhod Problem. II

  • Paul Dupuis1 &
  • Kavita Ramanan2 

Probability Theory and Related Fields volume 115, pages 197–236 (1999)Cite this article

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  • 28 Citations

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Abstract.

In this paper we consider Skorokhod Problems on polyhedral domains with a constant and possibly oblique constraint direction specified on each face of the domain, and with a corresponding cone of constraint directions at the intersection of faces. In part one of this paper we used convex duality to develop new methods for the construction of solutions to such Skorokhod Problems, and for proving Lipschitz continuity of the associated Skorokhod Maps. The main alternative approach to Skorokhod Problems of this type is the reflection mapping technique introduced by Harrison and Reiman [8]. In this part of the paper we apply the theory developed in part one to show that the reflection mapping technique of [8] is restricted to a slight generalization of the class of problems originally considered in [8]. We further illustrate the power of the duality approach by applying it to two other classes of Skorokhod Problems – those with normal directions of constraint, and a new class that arises from a model of processor sharing in communication networks. In particular, we prove existence of solutions to and Lipschitz continuity of the Skorokhod Maps associated with each of these Skorokhod Problems.

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Authors and Affiliations

  1. Lefschetz Center for Dynamical Systems, Division of Applied Mathematics, Brown University, Providence, RI 02912, USA (e-mail: dupuis@cfm.brown.edu), , , , , , US

    Paul Dupuis

  2. Bell Laboratories, Lucent Technologies, 600 Mountain Avenue, Murray Hill, New Jersey 07974, USA (e-mail: kavita@research.bell-labs.com), , , , , , US

    Kavita Ramanan

Authors
  1. Paul Dupuis
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  2. Kavita Ramanan
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Received: 17 April 1998 / Revised: 8 January 1999

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Dupuis, P., Ramanan, K. Convex duality and the Skorokhod Problem. II. Probab Theory Relat Fields 115, 197–236 (1999). https://doi.org/10.1007/s004400050270

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  • Issue Date: September 1999

  • DOI: https://doi.org/10.1007/s004400050270

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  • Mathematics Subject Classification (1991): 34A60, 52B11, 60K25, 60G99, 93A30
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