IEEE/CAA Journal of Automatica Sinica
Citation:  P. Huang, G. Wang, S. Wang, and H. Xiao, “A meanfield game for a forwardbackward stochastic system with partial observation and common noise,” IEEE/CAA J. Autom. Sinica, vol. 11, no. 3, pp. 746–759, Mar. 2024. doi: 10.1109/JAS.2023.124047 
This paper considers a linearquadratic (LQ) meanfield game governed by a forwardbackward stochastic system with partial observation and common noise, where a coupling structure enters state equations, cost functionals and observation equations. Firstly, to reduce the complexity of solving the meanfield game, a limiting control problem is introduced. By virtue of the decomposition approach, an admissible control set is proposed. Applying a filter technique and dimensionalexpansion technique, a decentralized control strategy and a consistency condition system are derived, and the related solvability is also addressed. Secondly, we discuss an approximate Nash equilibrium property of the decentralized control strategy. Finally, we work out a financial problem with some numerical simulations.
[1] 
J. M. Lasry and P. L. Lions, “Mean field games,” Jpn. J. Math., vol. 2, pp. 229–260, Mar. 2007. doi: 10.1007/s1153700706578

[2] 
M. Huang, P. E. Caines, and R. P. Malhamé, “Largepopulation costcoupled LQG problems with nonuniform agents: Individualmass behavior and decentralized εNash equilibria,” IEEE Trans. Autom. Control, vol. 52, pp. 1560–1571, Sept. 2007. doi: 10.1109/TAC.2007.904450

[3] 
R. Carmona, F. Delarue, and D. Lacker, “Mean field games with common noise,” Ann. Probab., vol. 44, no. 6, pp. 3740–3803, 2016.

[4] 
A. Aurell, R. Carmona, G. Dayanikli, and M. Lauriere, “Optimal incentives to mitigate epidemics: A Stackelberg mean field game approach,” SIAM J. Control Optim., vol. 60, no. 2, pp. 294–322, Apr. 2022. doi: 10.1137/20M1377862

[5] 
J. Moon and T. Bașar, “Linear quadratic mean field Stackelberg differential games,” Automatica, vol. 97, pp. 200–213, Dec. 2018. doi: 10.1016/j.automatica.2018.08.008

[6] 
A. Bensoussan, K. C. Sung, S. C. Yam, and S. P. Yung, “Linearquadratic mean field games,” J. Optim. Theory Appl., vol. 169, no. 2, pp. 496–529, 2016. doi: 10.1007/s1095701508194

[7] 
J. Moon and T. Bașar, “Linear quadratic risksensitive and robust mean field games,” IEEE Trans. Autom. Control, vol. 62, no. 3, pp. 1062–1077, Mar. 2017. doi: 10.1109/TAC.2016.2579264

[8] 
R. Xu and F. Zhang, “ϵNash meanfield games for general linearquadratic systems with applications,” Automatica, vol. 114, p. 108835, Jan. 2020.

[9] 
M. Li, N. Li, and Z. Wu, “Linearquadratic meanfield game for stochastic largepopulation systems with jump diffusion,” IET Control Theory Appl., vol. 14, no. 3, pp. 481–489, Feb. 2020. doi: 10.1049/ietcta.2019.0270

[10] 
B. Wang and J. Zhang, “Distributed output feedback control of Markov jump multiagent systems,” Automatica, vol. 49, no. 5, pp. 1397–1402, May 2013. doi: 10.1016/j.automatica.2013.01.063

[11] 
Y. Xu, Y. Yuan, Z. Wang, and X. Li, “Noncooperative model predictive game with Markov jump graph,” IEEE/CAA J. Autom. Sinica, vol. 10, no. 4, pp. 1–14, Apr. 2022.

[12] 
Y. Achdou, P. Mannucci, C. Marchi, and N. Tchou, “Deterministic mean field games with control on the acceleration and state constraints,” SIAM J. Math. Anal., vol. 54, no. 3, pp. 3757–3788, Jun. 2022. doi: 10.1137/21M1415492

[13] 
Y. Hu, J. Huang, and T. Nie, “LinearquadraticGaussian mixed meanfield games with heterogeneous input constraints,” SIAM J. Control Optim., vol. 56, no. 4, pp. 2835–2877, 2018. doi: 10.1137/17M1151420

[14] 
J. F. Bonnans, J. Gianatti, and L. Pfeiffer, “A Lagrangian approach for aggregative mean field games of controls with mixed and final constraints,” SIAM J. Control Optim., vol. 61, no. 1, pp. 105–134, Feb. 2023. doi: 10.1137/21M1407720

[15] 
M. Huang, P. Caines, and R. Malhamé, “Social optima in mean field LQG control: Centralized and decentralized strategies,” IEEE Trans. Autom. Control, vol. 57, no. 7, pp. 1736–1751, Jul. 2012. doi: 10.1109/TAC.2012.2183439

[16] 
B. Wang, H. Zhang, and J. Zhang, “Mean field linear quadratic control: Uniform stabilization and social optimality,” Automatica, vol. 121, pp. 1–14, Nov. 2020.

[17] 
K. Du, J. Huang, and Z. Wu, “Linear quadratic meanfieldgame of backward stochastic differential systems,” Math. Control Relat. Fields, vol. 8, pp. 653–678, 2018. doi: 10.3934/mcrf.2018028

[18] 
J. Huang, S. Wang, and Z. Wu, “Backward meanfield linearquadraticGaussian (LQG) games: Full and partial information,” IEEE Trans. Autom. Control, vol. 61, no. 12, pp. 3784–3796, Dec. 2016. doi: 10.1109/TAC.2016.2519501

[19] 
M. Ye, D. Li, Q.L. Han, and L. Ding, “Distributed Nash equilibrium seeking for general networked games with bounded disturbances,” IEEE/CAA J. Autom. Sinica, vol. 10, no. 2, pp. 376–387, Feb. 2023. doi: 10.1109/JAS.2022.105428

[20] 
B. Djehiche, H. Tembine, and R. Tempone, “A stochastic maximum principle for risksensitive meanfield type control,” IEEE Trans. Autom. Control, vol. 60, no. 10, pp. 2640–2649, Feb. 2015. doi: 10.1109/TAC.2015.2406973

[21] 
M. Hafayed and S. Meherrem, “On optimal control of meanfield stochastic systems driven by Teugels martingales via derivative with respect to measures,” Internat. J. Control, vol. 93, no. 5, pp. 1053–1062, 2020. doi: 10.1080/00207179.2018.1489148

[22] 
Y. Ni, J. Zhang, and M. Krstic, “Timeinconsistent meanfield stochastic LQ problem: Openloop timeconsistent control,” IEEE Trans. Autom. Control, vol. 63, no. 9, pp. 2771–2786, Sept. 2018. doi: 10.1109/TAC.2017.2776740

[23] 
B. Yang and J. Wu, “Infinite horizon optimal control for meanfield stochastic delay systems driven by Teugels martingales under partial information,” Optim. Control Appl. Meth., vol. 41, pp. 1371–1397, 2020. doi: 10.1002/oca.2602

[24] 
J. Yong, “A linearquadratic optimal control problem for meanfield stochastic differential equations,” SIAM J. Control Optim., vol. 51, no. 4, pp. 2809–2838, 2013. doi: 10.1137/120892477

[25] 
H. Wang, C. Zhang, K. Li, and X. Ma, “Game theorybased multiagent capacity optimization for integrated energy systems with compressed air energy storage,” Energy, vol. 221, p. 119777, Apr. 2021.

[26] 
S. Wu, “Partiallyobserved maximum principle for backward stochastic differential delay equations,” IEEE/CAA J. Autom. Sinica, pp. 1–6, Mar. 2017. DOI: 10.1109/JAS.2017.7510472.

[27] 
H. Ma and B. Liu, “Linearquadratic optimal control problem for partially observed forwardbackward stochastic differential equations of meanfield type,” Asian J. Control, vol. 18, no. 6, pp. 2146–2157, May 2016. doi: 10.1002/asjc.1310

[28] 
M. Wang, Q. Shi, and Q. Meng, “Optimal control of forwardbackward stochastic jumpdiffusion differential systems with observation noises: Stochastic maximum principle,” Asian J. Control, vol. 23, no. 1, pp. 241–254, 2021. doi: 10.1002/asjc.2272

[29] 
Y. Wang and L. Wang, “Forwardbackward stochastic differential games for optimal investment and dividend problem of an insurer under model uncertainty,” Appl. Math. Model., vol. 58, pp. 254–269, Jun. 2018. doi: 10.1016/j.apm.2017.07.027

[30] 
S. Zhang, J. Xiong, and X. Liu, “Stochastic maximum principle for partially observed forwardbackward stochastic differential equations with jumps and regime switching,” Sci. China Inf. Sci., vol. 61, no. 7, p. 070211, Jul. 2018. doi: 10.1007/s1143201792670

[31] 
S. Wang and H. Xiao, “Individual and mass behavior in large population forwardbackward stochastic control problems: Centralized and Nash equilibrium solutions,” Optim. Control Appl. Meth., vol. 42, pp. 1269–1292, Apr. 2021. doi: 10.1002/oca.2727

[32] 
A. Bensoussan, X. Feng, and J. Huang, “LinearquadraticGaussian meanfieldgame with partial observation and common noise,” Math. Control Relat. Fields, vol. 11, no. 1, pp. 23–46, 2021. doi: 10.3934/mcrf.2020025

[33] 
P. Huang, G. Wang, W. Wang, and Y. Wang, “A linearquadratic meanfield game of backward stochastic differential equation with partial information and common noise,” Appl. Math. Comput., vol. 446, p. 127899, 2023.

[34] 
G. Wang, Z. Wu, and J. Xiong, “A linearquadratic optimal control problem of forwardbackward stochastic differential equations with partial information,” IEEE Trans. Autom. Control, vol. 60, no. 11, pp. 2904–2916, Nov. 2015. doi: 10.1109/TAC.2015.2411871

[35] 
G. Wang, Z. Wu, and J. Xiong, An Introduction to Optimal Control of FBSDE With Incomplete Information. New York: SpringerVerlag, 2018.

[36] 
A. E. B. Lim and X. Zhou, “Linearquadratic control of backward stochatic differential equations,” SIAM J. Control Optim., vol. 40, no. 2, pp. 450–474, 2001. doi: 10.1137/S0363012900374737

[37] 
E. Pardoux, and S. Peng, “Adapted solution of backward stochastic equation,” Systems Control Lett., vol. 14, no. 1, pp. 55–61, Jan. 1990. doi: 10.1016/01676911(90)900826

[38] 
D. Majerek, W. Nowak, and W. Zieba, “Conditional strong law of large number,” Int. J. Pure Appl. Math., vol. 20, no. 2, pp. 143–156, 2005.

[39] 
J. Engwerda, LQ Dynamic Optimization and Differential Games. Chichester, West Sussex, England: John Wiley and Sons Ltd, 2005.
