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Weighted solutions of random time horizon BSDEs with stochastic monotonicity and general growth generators and related PDEs

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  • Li, Xinying
  • Zhang, Yaqi
  • Fan, Shengjun

Abstract

This study focuses on a multidimensional backward stochastic differential equation (BSDE) with a general random terminal time τ taking values in [0,+∞]. The generator g satisfies a stochastic monotonicity condition in the first unknown variable y and a stochastic Lipschitz continuity condition in the second unknown variable z, and it can have a more general growth with respect to y than the classical one stated in (H5) of Briand et al. (2003). Without imposing any restriction of finite moment on the stochastic coefficients, we establish a general existence and uniqueness result for the weighted solution of such BSDE in a proper weighted L2-space with a suitable weighted factor. This result is proved via some innovative ideas and delicate analytical techniques, and it unifies and strengthens some existing works on BSDEs with stochastic monotonicity generators, BSDEs with stochastic Lipschitz generators, and BSDEs with deterministic Lipschitz/monotonicity generators. Then, a continuous dependence property and a stability theorem for the weighted L2-solutions are given. We also derive the nonlinear Feynman–Kac formulas for both parabolic and elliptic PDEs in our context.

Suggested Citation

  • Li, Xinying & Zhang, Yaqi & Fan, Shengjun, 2025. "Weighted solutions of random time horizon BSDEs with stochastic monotonicity and general growth generators and related PDEs," Stochastic Processes and their Applications, Elsevier, vol. 190(C).
    • Handle: RePEc:eee:spapps:v:190:y:2025:i:c:s0304414925002029
    DOI: 10.1016/j.spa.2025.104758
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    References listed on IDEAS

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    1. N. El Karoui & S. Peng & M. C. Quenez, 1997. "Backward Stochastic Differential Equations in Finance," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 1-71, January.
    2. Fan, Shengjun & Hu, Ying, 2021. "Well-posedness of scalar BSDEs with sub-quadratic generators and related PDEs," Stochastic Processes and their Applications, Elsevier, vol. 131(C), pages 21-50.
    3. Ren, Yao-Feng, 2008. "On the Burkholder-Davis-Gundy inequalities for continuous martingales," Statistics & Probability Letters, Elsevier, vol. 78(17), pages 3034-3039, December.
    4. Briand, Philippe & Confortola, Fulvia, 2008. "BSDEs with stochastic Lipschitz condition and quadratic PDEs in Hilbert spaces," Stochastic Processes and their Applications, Elsevier, vol. 118(5), pages 818-838, May.
    5. Fan, ShengJun, 2016. "Bounded solutions, Lp(p>1) solutions and L1 solutions for one dimensional BSDEs under general assumptions," Stochastic Processes and their Applications, Elsevier, vol. 126(5), pages 1511-1552.
    6. Khaled Bahlali & El Hassan Essaky & Boubakeur Labed, 2004. "Reflected Backward Stochastic Differential Equation with Super-Linear Growth," Springer Books, in: Sergio Albeverio & Anne Boutet de Monvel & Habib Ouerdiane (ed.), Proceedings of the International Conference on Stochastic Analysis and Applications, pages 199-216, Springer.
    7. Jiajie Wang & Qikang Ran & Qihong Chen, 2007. "L p Solutions of BSDEs with Stochastic Lipschitz Condition," International Journal of Stochastic Analysis, Hindawi, vol. 2007, pages 1-14, March.
    8. Mao, Xuerong, 1995. "Adapted solutions of backward stochastic differential equations with non-Lipschitz coefficients," Stochastic Processes and their Applications, Elsevier, vol. 58(2), pages 281-292, August.
    9. K. Bahlali & A. Elouaflin & M. N'zi, 2004. "Backward stochastic differential equations with stochastic monotone coefficients," International Journal of Stochastic Analysis, Hindawi, vol. 2004, pages 1-19, January.
    10. Bender, Christian & Kohlmann, Michael, 2000. "BSDES With Stochastic Lipschitz Condition," CoFE Discussion Papers 00/08, University of Konstanz, Center of Finance and Econometrics (CoFE).
    11. 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.
    12. Xinying Li & Shengjun Fan, 2023. "Lp solutions of general time interval BSDEs with generators satisfying a p-order weak stochastic-monotonicity condition," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 52(16), pages 5650-5676, August.
    13. Owo, Jean-Marc, 2015. "Backward doubly stochastic differential equations with stochastic Lipschitz condition," Statistics & Probability Letters, Elsevier, vol. 96(C), pages 75-84.
    14. Ying Hu & Peter Imkeller & Matthias Muller, 2005. "Utility maximization in incomplete markets," Papers math/0508448, arXiv.org.
    15. Fan, ShengJun & Jiang, Long & Tian, DeJian, 2011. "One-dimensional BSDEs with finite and infinite time horizons," Stochastic Processes and their Applications, Elsevier, vol. 121(3), pages 427-440, March.
    16. K. Bahlali & B. Mezerdi & Y. Ouknine, 2004. "Prevalence of backward stochastic differential equations with unique solution," International Journal of Stochastic Analysis, Hindawi, vol. 2004, pages 1-14, January.
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    1. Xinman Cheng & Guanxing Fu & Xiaonyu Xia, 2026. "Stochastic Control Problems with Infinite Horizon and Regime Switching Arising in Optimal Liquidation with Semimartingale Strategies," Papers 2602.20552, arXiv.org, revised Feb 2026.

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