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Reflected BSDEs on filtered probability spaces

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  • Klimsiak, Tomasz

Abstract

We study the problem of existence and uniqueness of solutions of backward stochastic differential equations with two reflecting irregular barriers, Lp data and generators satisfying weak integrability conditions. We deal with equations on general filtered probability spaces. In case the generator does not depend on the z variable, we first consider the case p=1 and we only assume that the underlying filtration satisfies the usual conditions of right-continuity and completeness. Additional integrability properties of solutions are established if p∈(1,2] and the filtration is quasi-continuous. In case the generator depends on z, we assume that p=2, the filtration satisfies the usual conditions and additionally that it is separable. Our results apply for instance to Markov-type reflected backward equations driven by general Hunt processes.

Suggested Citation

  • Klimsiak, Tomasz, 2015. "Reflected BSDEs on filtered probability spaces," Stochastic Processes and their Applications, Elsevier, vol. 125(11), pages 4204-4241.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:11:p:4204-4241
    DOI: 10.1016/j.spa.2015.06.006
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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. Briand, Ph. & Delyon, B. & Hu, Y. & Pardoux, E. & Stoica, L., 2003. "Lp solutions of backward stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 108(1), pages 109-129, November.
    3. Nualart, David & Schoutens, Wim, 2000. "Chaotic and predictable representations for Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 90(1), pages 109-122, November.
    4. 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.
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    Cited by:

    1. Dumitrescu, Roxana & Elie, Romuald & Sabbagh, Wissal & Zhou, Chao, 2023. "A new Mertens decomposition of Yg,ξ-submartingale systems. Application to BSDEs with weak constraints at stopping times," Stochastic Processes and their Applications, Elsevier, vol. 164(C), pages 183-205.
    2. Eddahbi, M’hamed & Fakhouri, Imade & Ouknine, Youssef, 2020. "Reflected BSDEs with jumps in time-dependent convex càdlàg domains," Stochastic Processes and their Applications, Elsevier, vol. 130(11), pages 6515-6555.
    3. Yao, Song, 2017. "Lp solutions of backward stochastic differential equations with jumps," Stochastic Processes and their Applications, Elsevier, vol. 127(11), pages 3465-3511.
    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. Klimsiak, Tomasz & Rzymowski, Maurycy, 2023. "Nonlinear BSDEs on a general filtration with drivers depending on the martingale part of the solution," Stochastic Processes and their Applications, Elsevier, vol. 161(C), pages 424-450.
    6. Tianyang Nie & Edward Kim & Marek Rutkowski, 2018. "Arbitrage-Free Pricing of Game Options in Nonlinear Markets," Papers 1807.05448, arXiv.org.
    7. Safa Alsheyab & Tahir Choulli, 2021. "Reflected backward stochastic differential equations under stopping with an arbitrary random time," Papers 2107.11896, arXiv.org.
    8. T. Choulli & S. Alsheyab, 2024. "Linear reflected backward stochastic differential equations arising from vulnerable claims in markets with random horizon," Papers 2408.04758, arXiv.org.
    9. 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.
    10. Kim, Mun-Chol & O, Hun, 2021. "A general comparison theorem for reflected BSDEs," Statistics & Probability Letters, Elsevier, vol. 173(C).
    11. Bensoussan, Alain & Li, Yiqun & Yam, Sheung Chi Phillip, 2018. "Backward stochastic dynamics with a subdifferential operator and non-local parabolic variational inequalities," Stochastic Processes and their Applications, Elsevier, vol. 128(2), pages 644-688.

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    More about this item

    Keywords

    Reflected BSDE; General filtration; L1 data;
    All these keywords.

    JEL classification:

    • L1 - Industrial Organization - - Market Structure, Firm Strategy, and Market Performance

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