General Mean-Field BDSDEs with Continuous and Stochastic Linear Growth Coefficients

WANG Jinghan, SHI Yufeng, ZHAO Nana

Journal of Systems Science & Complexity ›› 2024, Vol. 37 ›› Issue (5) : 1887-1906.

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Journal of Systems Science & Complexity ›› 2024, Vol. 37 ›› Issue (5) : 1887-1906. DOI: 10.1007/s11424-024-3191-3

General Mean-Field BDSDEs with Continuous and Stochastic Linear Growth Coefficients

  • WANG Jinghan1, SHI Yufeng1, ZHAO Nana2
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Abstract

In this paper, the authors study a class of general mean-field BDSDEs whose coefficients satisfy some stochastic conditions. Specifically, the authors prove the existence and uniqueness theorem of solution under stochastic Lipschitz condition and obtain the related comparison theorem. Besides, the authors further relax the conditions and deduce the existence theorem of solutions under stochastic linear growth and continuous conditions, and the authors also prove the associated comparison theorem. Finally, an asset pricing problem is discussed, which demonstrates the application of the general mean-field BDSDEs in finance.

Key words

Backward doubly stochastic differential equations / comparison theorem / mean-field / stochastic conditions / Wasserstein metric

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WANG Jinghan , SHI Yufeng , ZHAO Nana. General Mean-Field BDSDEs with Continuous and Stochastic Linear Growth Coefficients. Journal of Systems Science & Complexity, 2024, 37(5): 1887-1906 https://doi.org/10.1007/s11424-024-3191-3

References

[1] Pardoux E and Peng S G, Adapted solution of a backward stochastic differential equation, Systems and Control Letters, 1990, 14(1): 55-61.
[2] El Karoui N, Peng S G, and Quenez M C, Backward stochastic differential equations in finance, Mathematical Finance, 1997, 7(1): 1-71.
[3] Shi J T and Wu Z, Maximum principle for forward-backward stochastic control system with random jumps and applications to finance, Journal of Systems Science & Complexity, 2010, 23(2): 219-231.
[4] Buckdahn R, Li J, Peng S G, et al., Mean-field stochastic differential equations and associated PDEs, Stochastic Analysis and Applications, 2017, 45(2): 824-878.
[5] Pardoux E, BSDEs, weak convergence and homogenization of semilinear PDEs, Nonlinear Analysis, Differential Equations and Control, 1999, 528: 503-549.
[6] Liu F and Peng S G, On controllability for stochastic control systems when the coefficient is time-variant, Journal of Systems Science & Complexity, 2010, 23(2): 270-278.
[7] Liu R Y and Wu Z, Well-posedness of fully coupled linear forward-backward stochastic differential equations, Journal of Systems Science & Complexity, 2019, 32(3): 789-802.
[8] Pardoux E and Peng S G, Backward doubly stochastic differential equations and systems of quasilinear SPDEs, Probability Theory and Related Fields, 1994, 98: 209-227.
[9] Shi Y F, Gu Y L, and Liu K, Comparison theorems of backward doubly stochastic differential equations and applications, Stochastic Analysis and Applications, 2005, 23: 97-110.
[10] Owo J M, Backward doubly stochastic differential equations with stochastic Lipschitz condition, Statistics and Probability Letters, 2015, 96(2015): 75-84.
[11] Owo J M, Backward doubly SDEs with continuous and stochastic linear growth coefficients, Random Operators and Stochastic Equations, 2018, 26(3): 175-184.
[12] Kac M, Foundations of kinetic theory, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1956, 3: 171-197.
[13] Lasry J and Lions P, Mean field games, Japanese Journal of Mathematics, 2007, 2: 229-260.
[14] Buckdahn R, Djehiche B, Li J, et al., Mean-field backward stochastic differential equations: A limit approach, The Annals of Probability, 2009, 37(4): 1524-1565.
[15] Buckdahn R, Li J, and Peng S G, Mean-field backward stochastic differential equations and related partial differential equations, Stochastic Processes and Their Applications, 2009, 119: 3133-3154.
[16] Li J, Liang H, and Zhang X, General mean-field BSDEs with continuous coefficients, Journal of Mathematical Analysis and Applications, 2018, 466: 264-280.
[17] Li J and Xing C Z, General mean-field BDSDEs with continuous coefficients, Journal of Mathematical Analysis and Applications, 2022, 506(2022): 125699.
[18] Zhao N N, Wang J H, Shi Y F, et al., General time-symmetric mean-field forward-backward doubly stochastic differential equations, Symmetry, 2023, 15(6): 1-24.
[19] Teng B, Applications of deep learning in numerical methods for stochastic differential equations and financial asset pricing, Ph.D Dissertation, Shandong University, Jinan, 2022(in Chinese).

Funding

This research was supported by the Zhiyuan Science Foundation of BIPT under Grant No. 2024212, National Key R&D Program of China under Grant No. 2018YFA0703900, the National Natural Science Foundation of China under Grant Nos. 11871309 and 11371226, and Natural Science Foundation of Shandong Province under Grant No. ZR2020QA026.
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