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Reflected Backward Stochastic Differential Equations Associated to Jump Markov Processes and Application to Partial Differential Equations

Author

Listed:
  • Abdelkarim Oualaid

    (LIBMA, Cadi Ayyad University)

  • Khaled Bahlali

    (Toulon University, IMATH)

  • Youssef Ouknine

    (LIBMA, Cadi Ayyad University
    Mohammed VI Polytechnic University)

Abstract

In this paper, we study a class of reflected backward stochastic differential equations (RBSDEs) driven by the compensated random measure associated to a given pure jump Markov process X on a general state space U. The reflection keeps the solution above a given càdlàg process. We prove the uniqueness and existence both by a combination of the Snell envelope theory and a fixed-point argument. As a consequence of these results, the Markovian structure of X allows us to represent probabilistically the solution of some quasi-variational inequalities problem.

Suggested Citation

  • Abdelkarim Oualaid & Khaled Bahlali & Youssef Ouknine, 2023. "Reflected Backward Stochastic Differential Equations Associated to Jump Markov Processes and Application to Partial Differential Equations," Journal of Theoretical Probability, Springer, vol. 36(3), pages 1400-1436, September.
    • Handle: RePEc:spr:jotpro:v:36:y:2023:i:3:d:10.1007_s10959-022-01212-x
    DOI: 10.1007/s10959-022-01212-x
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    References listed on IDEAS

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    1. Miryana Grigorova & Peter Imkeller & Elias Offen & Youssef Ouknine & Marie-Claire Quenez, 2017. "Reflected BSDEs when the obstacle is not right-continuous and optimal stopping," Post-Print hal-01141801, HAL.
    2. N. El Karoui & S. Peng & M. C. Quenez, 1997. "Backward Stochastic Differential Equations in Finance," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 1-71, January.
    3. Miryana Grigorova & Peter Imkeller & Elias Offen & Youssef Ouknine & Marie-Claire Quenez, 2015. "Reflected BSDEs when the obstacle is not right-continuous and optimal stopping," Papers 1504.06094, arXiv.org, revised May 2017.
    4. Royer, Manuela, 2006. "Backward stochastic differential equations with jumps and related non-linear expectations," Stochastic Processes and their Applications, Elsevier, vol. 116(10), pages 1358-1376, October.
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