Linear-Quadratic Pareto Cooperative Game for Mean-Field Backward Stochastic System

WANG Yu

系统科学与复杂性(英文) ›› 2024, Vol. 37 ›› Issue (3) : 947-964.

PDF(345 KB)
PDF(345 KB)
系统科学与复杂性(英文) ›› 2024, Vol. 37 ›› Issue (3) : 947-964. DOI: 10.1007/s11424-024-3091-6

Linear-Quadratic Pareto Cooperative Game for Mean-Field Backward Stochastic System

    WANG Yu
作者信息 +

Linear-Quadratic Pareto Cooperative Game for Mean-Field Backward Stochastic System

    WANG Yu
Author information +
文章历史 +

摘要

This paper focuses on a Pareto cooperative differential game with a linear mean-field backward stochastic system and a quadratic form cost functional. Based on a weighted sum optimality method, the Pareto game is equivalently converted to an optimal control problem. In the first place, the feedback form of Pareto optimal strategy is derived by virtue of decoupling technology, which is represented by four Riccati equations, a mean-field forward stochastic differential equation (MF-FSDE), and a mean-field backward stochastic differential equation (MF-BSDE). In addition, the corresponding Pareto optimal solution is further obtained. Finally, the author solves a problem in mathematical finance to illustrate the application of the theoretical results.

Abstract

This paper focuses on a Pareto cooperative differential game with a linear mean-field backward stochastic system and a quadratic form cost functional. Based on a weighted sum optimality method, the Pareto game is equivalently converted to an optimal control problem. In the first place, the feedback form of Pareto optimal strategy is derived by virtue of decoupling technology, which is represented by four Riccati equations, a mean-field forward stochastic differential equation (MF-FSDE), and a mean-field backward stochastic differential equation (MF-BSDE). In addition, the corresponding Pareto optimal solution is further obtained. Finally, the author solves a problem in mathematical finance to illustrate the application of the theoretical results.

关键词

Backward stochastic differential equation / linear-quadratic control / mean-field / Pareto optimality

Key words

Backward stochastic differential equation / linear-quadratic control / mean-field / Pareto optimality

引用本文

导出引用
WANG Yu. Linear-Quadratic Pareto Cooperative Game for Mean-Field Backward Stochastic System. 系统科学与复杂性(英文), 2024, 37(3): 947-964 https://doi.org/10.1007/s11424-024-3091-6
WANG Yu. Linear-Quadratic Pareto Cooperative Game for Mean-Field Backward Stochastic System. Journal of Systems Science and Complexity, 2024, 37(3): 947-964 https://doi.org/10.1007/s11424-024-3091-6

参考文献

[1] Liu X, Dong M X, Ota K, et al., Service pricing decision in cyber-physical systems: Insights from game theory, IEEE Transactions on Services Computing, 2016, 9(2): 186–198.
[2] Li Y, Mu Y F, Yuan S, et al., The game theoretical approach for multi-phase complex systems in chemical engineering, Journal of Systems Science & Complexity, 2017, 30(1): 4–19.
[3] Gao H W, Petrosyan L A, Qiao H, et al., Cooperation in two-stage games on undirected networks, Journal of Systems Science & Complexity, 2017, 30(3): 680–693.
[4] Chen B S, Chen W Y, Yang C T, et al., Noncooperative game strategy in cyber-financial systems with Wiener and Poisson random fluctuations: LMIs-constrained MOEA approach, IEEE Transactions on Cybernetics, 2018, 48(12): 3323–3336.
[5] Engwerda J C, The regular convex cooperative linear quadratic control problem, Automatica, 2008, 44(9): 2453–2457.
[6] Engwerda J C, Necessary and sufficient conditions for Pareto optimal solutions of cooperative differential games, SIAM Journal on Control and Optimization, 2010, 48(6): 3859–3881.
[7] Huang Y B and Zhao J, Pareto efficiency of finite horizon switched linear quadratic differential games, Journal of Systems Science & Complexity, 2018, 31(1): 173–187.
[8] Lin Y N, Jiang X S, and Zhang W H, Necessary and sufficient conditions for Pareto optimality of the stochastic systems in finite horizon, Automatica, 2018, 94: 341–348.
[9] Yeung D W K and Petrosyan L A, Cooperative dynamic games with control lags, Dynamic Games and Applications, 2019, 9(2): 550–567.
[10] Lin C and Chen B S, Achieving Pareto optimal power tracking control for interference limited wireless systems via multi-objective H2/H optimization, IEEE Transactions on Wireless Communications, 2013, 12(12): 6154–6165.
[11] Yeung D W K and Petrosyan L A, Subgame consistent cooperative solutions in stochastic differential games, Journal of Optimization Theory and Applications, 2004, 120(3): 651–666.
[12] Yeung D W K and Petrosyan L A, Cooperative Stochastic Differential Games, Springer, New York, 2006.
[13] Yeung D W K and Petrosyan L A, Subgame consistent cooperative solution of dynamic games with random horizon, Journal of Optimization Theory and Applications, 2011, 150(1): 78–97.
[14] Kac M, Foundations of kinetic theory, Berkeley and Los Angeles, University of California Press, California, 1956, 3(600): 171–197.
[15] Talay D and Vaillant O, A stochastic particle method with random weights for the computation of statistical solutions of McKean-Vlasov equations, The Annals of Applied Probability, 2003, 13(1): 140–180.
[16] Lasry J M and Lions P L, Mean field games, Japanese Journal of Mathematics, 2007, 2(1): 229–260.
[17] Li C L, Liu Z M, Wu J B, et al., The stochastic maximum principle for a jump-diffusion meanfield model involving impulse controls and applications in finance, Journal of Systems Science & Complexity, 2020, 33(1): 26–42.
[18] Moon J, Linear-quadratic mean field stochastic zero-sum differential games, Automatica, 2020, 120: 109067.
[19] Li M and Wu Z, Linear-quadratic non-zero sum differential game for mean-field stochastic systems with asymmetric information, Journal of Mathematical Analysis and Applications, 2021, 504: 125315.
[20] Wang G C and Zhang S S, A mean-field linear-quadratic stochastic Stackelberg differential game with one leader and two followers, Journal of Systems Science & Complexity, 2020, 33(5): 1383– 1401.
[21] Wang B C and Zhang J F, Social optima in mean field linear-quadratic-Gaussian models with Markov jump parameters, SIAM Journal on Control and Optimization, 2017, 55(1): 429–456.
[22] Lin Y N, Necessary/sufficient conditions for Pareto optimality in finite horizon mean-field type stochastic differential game, Automatica, 2020, 119: 108951.
[23] Lin Y N and Zhang W H, Pareto efficiency in the infinite horizon mean-field type cooperative stochastic differential game, Journal of the Franklin Institute, 2021, 358(10): 5532–5551.
[24] Wang B C and Zhang H S, Indefinite linear quadratic mean field social control problems with multiplicative noise, IEEE Transactions on Automatic Control, 2021, 66(11): 5221–5236.
[25] Bismut J M, An introductory approach to duality in optimal stochastic control, SIAM Review, 1978, 20(1): 62–78.
[26] Pardoux E and Peng S G, Adapted solution of backward stochastic differential equation, Systems Control Letters, 1990, 14(1): 55–61.
[27] 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.
[28] Wang G C and Yu Z Y, A partial information non-zero sum differential game of backward stochastic differential equations with applications, Automatica, 2012, 48(2): 342–352.
[29] Wang G C, Xiao H, and Xiong J, A kind of LQ non-zero sum differential game of backward stochastic differential equation with asymmetric information, Automatica, 2018, 97: 346–352.
[30] Du K and Wu Z, Linear-quadratic Stackelberg game for mean-field backward stochastic differential system and application, Mathematical Problems in Engineering, 2019, 2019: 1–17.
[31] Nie P P, Wang G C, and Wang Y, Necessary and sufficient conditions for Pareto optimal solution of backward stochastic system with application, IEEE Transactions on Automatic Control, 2023, 68(11): 6696–6710.
[32] Wang Y, Pareto Cooperative Differential Game of Mean-Field Backward Stochastic System, Shandong University, MA thesis, Jinan, 2022, in Chinese.
[33] Engwerda J C, LQ Dynamic Optimization and Differential Games, John Wiley and Sons, New York, 2005.
[34] Li X, Sun J R, and Xiong J, Linear quadratic optimal control problems for mean-field backward stochastic differential equations, Applied Mathematics and Optimization, 2019, 80(1): 223–250.
[35] Lim A E B and Zhou X Y, Linear-quadratic control of backward stochastic differential equations, SIAM Journal on Control and Optimization, 2001, 40(2): 450–474.
[36] Zhang J F, A numerical scheme for BSDEs, The Annals of Applied Probability, 2004, 14(1): 459–488.

基金

This research was supported by the National Key R&D Program of China under Grant No. 2022YFA1006103, the National Natural Science Foundation of China under Grant Nos. 61821004, 61925306, and 11831010, and the Natural Science Foundation of Shandong Province under Grant Nos. ZR2019ZD42 and ZR2020ZD24.
PDF(345 KB)

110

Accesses

0

Citation

Detail

段落导航
相关文章

/