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Nonlinear BSDEs on a general filtration with drivers depending on the martingale part of the solution

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  • Klimsiak, Tomasz
  • Rzymowski, Maurycy

Abstract

In the present paper, we consider multidimensional nonlinear backward stochastic differential equations (BSDEs) with a driver depending on the martingale part M of a solution. We assume that the nonlinear term is merely monotone continuous with respect to the state variable. As to the regularity of the driver with respect to the martingale variable, we consider a very general condition which permits path-dependence on “the future” of the process M as well as a dependence of its law (McKean–Vlasov-type equations). For such drivers, we prove the existence and uniqueness of a global solution (i.e. for any maturity T>0) to a BSDE with data satisfying natural integrability conditions.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:161:y:2023:i:c:p:424-450
    DOI: 10.1016/j.spa.2023.04.011
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    References listed on IDEAS

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    1. Klimsiak, Tomasz, 2015. "Reflected BSDEs on filtered probability spaces," Stochastic Processes and their Applications, Elsevier, vol. 125(11), pages 4204-4241.
    2. 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.
    3. 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.
    4. Buckdahn, Rainer & Li, Juan & Peng, Shige, 2009. "Mean-field backward stochastic differential equations and related partial differential equations," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3133-3154, October.
    5. 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.
    6. Royer, Manuela, 2006. "Backward stochastic differential equations with jumps and related non-linear expectations," Stochastic Processes and their Applications, Elsevier, vol. 116(10), pages 1358-1376, October.
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