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Reflected and Doubly Reflected Backward Stochastic Differential Equations with Time-Delayed Generators

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  • Monia Karouf

    (Gabès University)

Abstract

In this paper, we study reflected backward stochastic differential equations with two reflecting barriers and time-delayed generators (delay RBSDEs for short). We consider the case of Brownian noise as well as the case of Brownian and Poisson noise. For both cases, we show the existence and uniqueness of the solution and give a comparison theorem.

Suggested Citation

  • Monia Karouf, 2019. "Reflected and Doubly Reflected Backward Stochastic Differential Equations with Time-Delayed Generators," Journal of Theoretical Probability, Springer, vol. 32(1), pages 216-248, March.
    • Handle: RePEc:spr:jotpro:v:32:y:2019:i:1:d:10.1007_s10959-018-0829-x
    DOI: 10.1007/s10959-018-0829-x
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    References listed on IDEAS

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    1. Li Chen & Jianhui Huang, 2015. "Stochastic Maximum Principle for Controlled Backward Delayed System via Advanced Stochastic Differential Equation," Journal of Optimization Theory and Applications, Springer, vol. 167(3), pages 1112-1135, December.
    2. Zhou, Qing & Ren, Yong, 2012. "Reflected backward stochastic differential equations with time delayed generators," Statistics & Probability Letters, Elsevier, vol. 82(5), pages 979-990.
    3. 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.
    4. Lifen An & Shaolin Ji, 2010. "Reflected Backward Stochastic Difference Equations with Finite State and their applications," Papers 1001.3054, arXiv.org, revised Dec 2012.
    5. 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.
    6. Briand, Philippe & Elie, Romuald, 2013. "A simple constructive approach to quadratic BSDEs with or without delay," Stochastic Processes and their Applications, Elsevier, vol. 123(8), pages 2921-2939.
    7. Delong, Lukasz & Imkeller, Peter, 2010. "On Malliavin's differentiability of BSDEs with time delayed generators driven by Brownian motions and Poisson random measures," Stochastic Processes and their Applications, Elsevier, vol. 120(9), pages 1748-1775, August.
    8. dos Reis, Gonçalo & Réveillac, Anthony & Zhang, Jianing, 2011. "FBSDEs with time delayed generators: Lp-solutions, differentiability, representation formulas and path regularity," Stochastic Processes and their Applications, Elsevier, vol. 121(9), pages 2114-2150, September.
    9. 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.
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