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Well-posedness of mean-field type forward–backward stochastic differential equations

Author

Listed:
  • Bensoussan, A.
  • Yam, S.C.P.
  • Zhang, Z.

Abstract

Being motivated by a recent pioneer work Carmona and Delarue (2013), in this article, we propose a broad class of natural monotonicity conditions under which the unique existence of the solutions to Mean-Field Type (MFT) Forward–Backward Stochastic Differential Equations (FBSDE) can be established. Our conditions provided here are consistent with those normally adopted in the traditional FBSDE (without the interference of a mean-field) frameworks, and give a generic explanation on the unique existence of solutions to common MFT-FBSDEs, such as those in the linear-quadratic setting; besides, the conditions are ‘optimal’ in a certain sense that can elaborate on how their counter-example in Carmona and Delarue (2013) just fails to ensure its well-posedness. Finally, a stability theorem is also included.

Suggested Citation

  • Bensoussan, A. & Yam, S.C.P. & Zhang, Z., 2015. "Well-posedness of mean-field type forward–backward stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 125(9), pages 3327-3354.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:9:p:3327-3354
    DOI: 10.1016/j.spa.2015.04.006
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    References listed on IDEAS

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    1. 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.
    2. Delarue, François, 2002. "On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case," Stochastic Processes and their Applications, Elsevier, vol. 99(2), pages 209-286, June.
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    Cited by:

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    2. Huang, Zhipeng & Negyesi, Balint & Oosterlee, Cornelis W., 2025. "Convergence of the deep BSDE method for stochastic control problems formulated through the stochastic maximum principle," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 227(C), pages 553-568.
    3. Tianjiao Hua & Peng Luo, 2025. "Well-Posedness for a Class of Mean-Field-Type Forward-Backward Stochastic Differential Equations and Classical Solutions of Related Master Equations," Journal of Theoretical Probability, Springer, vol. 38(1), pages 1-29, March.
    4. A. Bensoussan & K. C. J. Sung & S. C. P. Yam & S. P. Yung, 2016. "Linear-Quadratic Mean Field Games," Journal of Optimization Theory and Applications, Springer, vol. 169(2), pages 496-529, May.
    5. Ang Ke & Jinbiao Wu & Biteng Xu, 2025. "Optimal Strategy of Mean-Field FBSDE Games with Delay and Noisy Memory Based on Malliavin Calculus," Dynamic Games and Applications, Springer, vol. 15(3), pages 906-946, July.
    6. Ahuja, Saran & Ren, Weiluo & Yang, Tzu-Wei, 2019. "Forward–backward stochastic differential equations with monotone functionals and mean field games with common noise," Stochastic Processes and their Applications, Elsevier, vol. 129(10), pages 3859-3892.
    7. Sin, Myong-Guk & Ri, Kyong-Il & Kim, Kyong-Hui, 2022. "Existence and uniqueness of solution for coupled fractional mean-field forward–backward stochastic differential equations," Statistics & Probability Letters, Elsevier, vol. 190(C).
    8. Sun, Shengqiu, 2025. "Mean-field forward–backward stochastic differential equations driven by G-Brownian motion," Statistics & Probability Letters, Elsevier, vol. 223(C).
    9. René Carmona & Jean-Pierre Fouque & Seyyed Mostafa Mousavi & Li-Hsien Sun, 2018. "Systemic Risk and Stochastic Games with Delay," Journal of Optimization Theory and Applications, Springer, vol. 179(2), pages 366-399, November.

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