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The Skorokhod problem in a time-dependent interval

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  • Burdzy, Krzysztof
  • Kang, Weining
  • Ramanan, Kavita

Abstract

We consider the Skorokhod problem in a time-varying interval. We prove existence and uniqueness of the solution. We also express the solution in terms of an explicit formula. Moving boundaries may generate singularities when they touch. Under the assumption that the first time [tau] when the moving boundaries touch after time zero is strictly positive, we derive two sets of conditions on the moving boundaries. We show that the variation of the local time of the associated reflected Brownian motion on [0,[tau]] is finite under the first set of conditions and infinite under the second set of conditions. We also apply these results to study the semimartingale property of a class of two-dimensional reflected Brownian motions.

Suggested Citation

  • Burdzy, Krzysztof & Kang, Weining & Ramanan, Kavita, 2009. "The Skorokhod problem in a time-dependent interval," Stochastic Processes and their Applications, Elsevier, vol. 119(2), pages 428-452, February.
    • Handle: RePEc:eee:spapps:v:119:y:2009:i:2:p:428-452
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    1. Avi Mandelbaum & William A. Massey, 1995. "Strong Approximations for Time-Dependent Queues," Mathematics of Operations Research, INFORMS, vol. 20(1), pages 33-64, February.
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    4. Falkowski, Adrian & Słomiński, Leszek, 2021. "Mean reflected stochastic differential equations with two constraints," Stochastic Processes and their Applications, Elsevier, vol. 141(C), pages 172-196.
    5. Dianetti, Jodi & Ferrari, Giorgio, 2021. "Multidimensional Singular Control and Related Skorokhod Problem: Suficient Conditions for the Characterization of Optimal Controls," Center for Mathematical Economics Working Papers 645, Center for Mathematical Economics, Bielefeld University.
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    8. Atar, Rami & Biswas, Anup & Kaspi, Haya, 2018. "Law of large numbers for the many-server earliest-deadline-first queue," Stochastic Processes and their Applications, Elsevier, vol. 128(7), pages 2270-2296.
    9. Xin Guo & Wenpin Tang & Renyuan Xu, 2018. "A class of stochastic games and moving free boundary problems," Papers 1809.03459, arXiv.org, revised Oct 2021.
    10. Falkowski, Adrian & Słomiński, Leszek, 2017. "SDEs with constraints driven by semimartingales and processes with bounded p-variation," Stochastic Processes and their Applications, Elsevier, vol. 127(11), pages 3536-3557.
    11. Allan, Andrew L. & Liu, Chong & Prömel, David J., 2021. "Càdlàg rough differential equations with reflecting barriers," Stochastic Processes and their Applications, Elsevier, vol. 142(C), pages 79-104.
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    17. Maxim Bichuch, 2011. "Pricing a Contingent Claim Liability with Transaction Costs Using Asymptotic Analysis for Optimal Investment," Papers 1112.3012, arXiv.org.
    18. Hong Zhang & Saviour Worlanyo Akuamoah & Wilson Osafo Apeanti & Prince Harvim & David Yaro & Paul Georgescu, 2021. "The Stability Analysis of a Double-X Queuing Network Occurring in the Banking Sector," Mathematics, MDPI, vol. 9(16), pages 1-21, August.
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    20. Lee, Chihoon & Weerasinghe, Ananda, 2011. "Convergence of a queueing system in heavy traffic with general patience-time distributions," Stochastic Processes and their Applications, Elsevier, vol. 121(11), pages 2507-2552, November.

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