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Doubly Reflected BSDEs with Integrable Parameters and Related Dynkin Games

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  • Erhan Bayraktar
  • Song Yao

Abstract

We study a doubly reflected backward stochastic differential equation (BSDE) with integrable parameters and the related Dynkin game. When the lower obstacle $L$ and the upper obstacle $U$ of the equation are completely separated, we construct a unique solution of the doubly reflected BSDE by pasting local solutions and show that the $Y-$component of the unique solution represents the value process of the corresponding Dynkin game under $g-$evaluation, a nonlinear expectation induced by BSDEs with the same generator $g$ as the doubly reflected BSDE concerned. In particular, the first time when process $Y $ meets $L$ and the first time when process $Y $ meets $U$ form a saddle point of the Dynkin game.

Suggested Citation

  • Erhan Bayraktar & Song Yao, 2014. "Doubly Reflected BSDEs with Integrable Parameters and Related Dynkin Games," Papers 1412.2053, arXiv.org, revised Jul 2015.
    • Handle: RePEc:arx:papers:1412.2053
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    References listed on IDEAS

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    1. Bayraktar, Erhan & Yao, Song, 2011. "Optimal stopping for non-linear expectations--Part I," Stochastic Processes and their Applications, Elsevier, vol. 121(2), pages 185-211, February.
    2. Bayraktar, Erhan & Yao, Song, 2012. "Quadratic reflected BSDEs with unbounded obstacles," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1155-1203.
    3. Erhan Bayraktar & Song Yao, 2013. "On the Robust Optimal Stopping Problem," Papers 1301.0091, arXiv.org, revised Apr 2016.
    4. Erhan Bayraktar & Ioannis Karatzas & Song Yao, 2009. "Optimal Stopping for Dynamic Convex Risk Measures," Papers 0909.4948, arXiv.org, revised Nov 2009.
    5. Bahlali, Khaled & Hamadène, SaI¨d & Mezerdi, Brahim, 2005. "Backward stochastic differential equations with two reflecting barriers and continuous with quadratic growth coefficient," Stochastic Processes and their Applications, Elsevier, vol. 115(7), pages 1107-1129, July.
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    9. Erhan Bayraktar & Yu-Jui Huang, 2010. "On the Multi-Dimensional Controller and Stopper Games," Papers 1009.0932, arXiv.org, revised Jan 2013.
    10. Marcel Nutz & Jianfeng Zhang, 2012. "Optimal stopping under adverse nonlinear expectation and related games," Papers 1212.2140, arXiv.org, revised Sep 2015.
    11. Touzi, N. & Vieille, N., 1999. "Continuous-Time Dynkin Games with Mixed Strategies," Papiers d'Economie Mathématique et Applications 1999.112, Université Panthéon-Sorbonne (Paris 1).
    12. Erhan Bayraktar & Song Yao, 2009. "Optimal Stopping for Non-linear Expectations," Papers 0905.3601, arXiv.org, revised Jan 2011.
    13. Rosazza Gianin, Emanuela, 2006. "Risk measures via g-expectations," Insurance: Mathematics and Economics, Elsevier, vol. 39(1), pages 19-34, August.
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    Cited by:

    1. 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.
    2. Marzougue, Mohamed, 2023. "Non-linear Dynkin games over split stopping times," Statistics & Probability Letters, Elsevier, vol. 193(C).
    3. Essaky, E.H. & Hassani, M. & Rhazlane, C.E., 2023. "Doubly reflected BSDEs with stochastic quadratic growth: Around the predictable obstacles," Stochastic Processes and their Applications, Elsevier, vol. 163(C), pages 473-497.
    4. Tianyang Nie & Edward Kim & Marek Rutkowski, 2018. "Arbitrage-Free Pricing of Game Options in Nonlinear Markets," Papers 1807.05448, arXiv.org.
    5. Erhan Bayraktar & Song Yao, 2015. "On the Robust Dynkin Game," Papers 1506.09184, arXiv.org, revised Sep 2016.
    6. Siyu Lv & Zhen Wu & Qing Zhang, 2022. "The Dynkin game with regime switching and applications to pricing game options," Annals of Operations Research, Springer, vol. 313(2), pages 1159-1182, June.

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