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Non-semimartingale solutions of reflected BSDEs and applications to Dynkin games

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  • Klimsiak, Tomasz

Abstract

We introduce a new class of reflected backward stochastic differential equations with two càdlàg barriers, which need not satisfy any separation conditions. For that reason, in general, the solutions are not semimartingales. We prove existence, uniqueness and approximation results for solutions of equations defined on general filtered probability spaces. Applications to nonlinear Dynkin games are given.

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  • Klimsiak, Tomasz, 2021. "Non-semimartingale solutions of reflected BSDEs and applications to Dynkin games," Stochastic Processes and their Applications, Elsevier, vol. 134(C), pages 208-239.
    • Handle: RePEc:eee:spapps:v:134:y:2021:i:c:p:208-239
    DOI: 10.1016/j.spa.2020.12.008
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    References listed on IDEAS

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    1. Hamadène, S. & Wang, H., 2009. "BSDEs with two RCLL reflecting obstacles driven by Brownian motion and Poisson measure and a related mixed zero-sum game," Stochastic Processes and their Applications, Elsevier, vol. 119(9), pages 2881-2912, September.
    2. Klimsiak, Tomasz, 2015. "Reflected BSDEs on filtered probability spaces," Stochastic Processes and their Applications, Elsevier, vol. 125(11), pages 4204-4241.
    3. 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.
    4. Klimsiak, Tomasz, 2019. "Systems of quasi-variational inequalities related to the switching problem," Stochastic Processes and their Applications, Elsevier, vol. 129(4), pages 1259-1286.
    5. 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.
    6. Lin, Qian, 2015. "Nash equilibrium payoffs for stochastic differential games with jumps and coupled nonlinear cost functionals," Stochastic Processes and their Applications, Elsevier, vol. 125(12), pages 4405-4454.
    7. Bayraktar, Erhan & Yao, Song, 2015. "Doubly reflected BSDEs with integrable parameters and related Dynkin games," Stochastic Processes and their Applications, Elsevier, vol. 125(12), pages 4489-4542.
    8. 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.
    9. Rozkosz, Andrzej & Słomiński, Leszek, 2012. "Lp solutions of reflected BSDEs under monotonicity condition," Stochastic Processes and their Applications, Elsevier, vol. 122(12), pages 3875-3900.
    10. Miryana Grigorova & Peter Imkeller & Youssef Ouknine & Marie-Claire Quenez, 2018. "Doubly Reflected BSDEs and ${\cal E}^{f}$-Dynkin games: beyond the right-continuous case," Working Papers hal-01497914, HAL.
    11. Grigorova, Miryana & Imkeller, Peter & Quenez, Marie-Claire & Ouknine, Youssef, 2018. "Doubly Reflected BSDEs and $\mathcal{E}$$^ƒ$-Dynkin games: beyond the right-continuous case," Center for Mathematical Economics Working Papers 598, Center for Mathematical Economics, Bielefeld University.
    12. Klimsiak, Tomasz & Rzymowski, Maurycy & Słomiński, Leszek, 2019. "Reflected BSDEs with regulated trajectories," Stochastic Processes and their Applications, Elsevier, vol. 129(4), pages 1153-1184.
    13. Quenez, Marie-Claire & Sulem, Agnès, 2014. "Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps," Stochastic Processes and their Applications, Elsevier, vol. 124(9), pages 3031-3054.
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