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Reflected BSDEs with nonpositive jumps, and controller-and-stopper games

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  • Choukroun, Sébastien
  • Cosso, Andrea
  • Pham, Huyên

Abstract

We study a class of reflected backward stochastic differential equations with nonpositive jumps and upper barrier. Existence and uniqueness of a minimal solution are proved by a double penalization approach under regularity assumptions on the obstacle. In a suitable regime switching diffusion framework, we show the connection between our class of BSDEs and fully nonlinear variational inequalities. Our BSDE representation provides in particular a Feynman–Kac type formula for PDEs associated to general zero-sum stochastic differential controller-and-stopper games, where control affects both drift and diffusion term, and the diffusion coefficient can be degenerate. Moreover, we state a dual game formula of this BSDE minimal solution involving equivalent change of probability measures, and discount processes. This gives in particular a new representation for zero-sum stochastic differential controller-and-stopper games.

Suggested Citation

  • Choukroun, Sébastien & Cosso, Andrea & Pham, Huyên, 2015. "Reflected BSDEs with nonpositive jumps, and controller-and-stopper games," Stochastic Processes and their Applications, Elsevier, vol. 125(2), pages 597-633.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:2:p:597-633
    DOI: 10.1016/j.spa.2014.09.015
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    References listed on IDEAS

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    1. Kharroubi Idris & Langrené Nicolas & Pham Huyên, 2014. "A numerical algorithm for fully nonlinear HJB equations: An approach by control randomization," Monte Carlo Methods and Applications, De Gruyter, vol. 20(2), pages 145-165, June.
    2. Matoussi, Anis, 1997. "Reflected solutions of backward stochastic differential equations with continuous coefficient," Statistics & Probability Letters, Elsevier, vol. 34(4), pages 347-354, June.
    3. Lepeltier, J.-P. & Xu, M., 2005. "Penalization method for reflected backward stochastic differential equations with one r.c.l.l. barrier," Statistics & Probability Letters, Elsevier, vol. 75(1), pages 58-66, November.
    4. Hamadène, S. & Lepeltier, J. -P., 2000. "Reflected BSDEs and mixed game problem," Stochastic Processes and their Applications, Elsevier, vol. 85(2), pages 177-188, February.
    5. repec:dau:papers:123456789/12195 is not listed on IDEAS
    6. Quenez, Marie-Claire & Sulem, Agnès, 2013. "BSDEs with jumps, optimization and applications to dynamic risk measures," Stochastic Processes and their Applications, Elsevier, vol. 123(8), pages 3328-3357.
    7. 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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    Cited by:

    1. Perninge, Magnus, 2024. "Optimal stopping of BSDEs with constrained jumps and related zero-sum games," Stochastic Processes and their Applications, Elsevier, vol. 173(C).
    2. Bandini, Elena & Cosso, Andrea & Fuhrman, Marco & Pham, Huyên, 2019. "Randomized filtering and Bellman equation in Wasserstein space for partial observation control problem," Stochastic Processes and their Applications, Elsevier, vol. 129(2), pages 674-711.
    3. Miryana Grigorova & Marie-Claire Quenez & Agnès Sulem, 2019. "American options in a non-linear incomplete market model with default," Working Papers hal-02025835, HAL.
    4. Miryana Grigorova & Marie-Claire Quenez & Agnès Sulem, 2021. "American options in a non-linear incomplete market model with default," Post-Print hal-02025835, HAL.
    5. Grigorova, Miryana & Quenez, Marie-Claire & Sulem, Agnès, 2019. "Superhedging prices of European and American options in a non-linear incomplete market with default," Center for Mathematical Economics Working Papers 607, Center for Mathematical Economics, Bielefeld University.

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