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Reflected Solutions of BSDEs Driven by $\textit{G}$-Brownian Motion

Author

Listed:
  • Li, Hanwu

    (Center for Mathematical Economics, Bielefeld University)

  • Peng, Shige

    (Center for Mathematical Economics, Bielefeld University)

  • Soumana Hima, Abdoulaye

    (Center for Mathematical Economics, Bielefeld University)

Abstract

In this paper, we study the reflected solutions of one-dimensional backward stochastic differential equations driven by *G*-Brownian motion (RGBSDE for short). The reflection keeps the solution above a given stochastic process. In order to derive the uniqueness of reflected *G*-BSDEs, we apply a \martingale condition" instead of the Skorohod condition. Similar to the classical case, we prove the existence by approximation via penalization.

Suggested Citation

  • Li, Hanwu & Peng, Shige & Soumana Hima, Abdoulaye, 2018. "Reflected Solutions of BSDEs Driven by $\textit{G}$-Brownian Motion," Center for Mathematical Economics Working Papers 590, Center for Mathematical Economics, Bielefeld University.
    • Handle: RePEc:bie:wpaper:590
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    References listed on IDEAS

    as
    1. Matoussi, Anis, 1997. "Reflected solutions of backward stochastic differential equations with continuous coefficient," Statistics & Probability Letters, Elsevier, vol. 34(4), pages 347-354, June.
    2. 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.
    3. Hu, Mingshang & Ji, Shaolin & Peng, Shige & Song, Yongsheng, 2014. "Comparison theorem, Feynman–Kac formula and Girsanov transformation for BSDEs driven by G-Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 124(2), pages 1170-1195.
    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. Peng, Shige, 2008. "Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation," Stochastic Processes and their Applications, Elsevier, vol. 118(12), pages 2223-2253, December.
    6. Possamaï, Dylan, 2013. "Second order backward stochastic differential equations under a monotonicity condition," Stochastic Processes and their Applications, Elsevier, vol. 123(5), pages 1521-1545.
    7. Joerg Vorbrink, 2010. "Financial markets with volatility uncertainty," Papers 1012.1535, arXiv.org, revised Dec 2010.
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