A Variational Formula of Forward-Backward Stochastic Differential System of Mean-Field Type with Observation Noise and Some Application

doi:10.1007/s42967-024-00431-9

Abstract

Abstract: This paper examines an optimal control problem for mean-field systems under partial observation. The state system is described by a controlled mean-field forward-backward stochastic differential equation that features correlated noises between the system and the observation. Furthermore, the observation coefficients are allowed to depend not only on the control process but also on its probability distribution. Assuming a convex control domain and allowing all coefficients of the systems to be random, we derive directly a variational formula for the cost function in a given control process direction in terms of the Hamiltonian and the associated adjoint system without relying on variational systems under standard assumptions on the coefficients. As an application, we present the necessary and sufficient conditions for the optimality of our control problem using Pontryagin’s maximum principle in a unified manner. The research provides insights into the optimization of mean-field systems under partial observation, which has practical implications for various applications.

Key words: Mean-field, Forward-backward stochastic differential equation (FBSDE), Partial observation, Girsanov’s theorem, Maximum principle

CLC Number:

Meijiao Wang, Maoning Tang, Qiuhong Shi, Qingxin Meng. A Variational Formula of Forward-Backward Stochastic Differential System of Mean-Field Type with Observation Noise and Some Application[J]. Communications on Applied Mathematics and Computation, 2026, 8(1): 269-286.

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References

[1] Antonelli, F.: Backward-forward stochastic differential equations. Ann. Appl. Prob. 3, 777-793 (1993)
[2] Buckdahn, R., Djehiche, B., Li, J.: A general stochastic maximum principle for SDEs of mean-field type. Appl. Math. Optim. 64(2), 197-216 (2011)
[3] Delavarkhalafi, A., Fatemion Aghda, A.S., Tahmasebi, M.: Maximum principle for infinite horizon optimal control of mean-field backward stochastic systems with delay and noisy memory. Int. J. Control 95(2), 535-543 (2022)
[4] Du, K., Huang, J., Wu, Z.: Linear quadratic mean-field-game of backward stochastic differential systems. Math. Control Relat. Fields 8(3/4), 653 (2018)
[5] Duffie, D., Epstein, L.G.: Asset pricing with stochastic differential utility. Rev. Fin. Stud. 5(3), 411-436 (1992)
[6] El Karoui, N., Peng, S., Quenez, M.C.: A dynamic maximum principle for the optimization of recursive utilities under constraints. Ann. Appl. Prob. 11, 664-693 (2001)
[7] Fu, Q., Xu, F., Shen, T., Takai, K.: Distributed optimal energy consumption control of HEVs under MFG-based speed consensus. Control Theory Technol. 18(2), 193-203 (2020)
[8] Hafayed, M., Ghebouli, M., Boukaf, S., Shi, Y.: Partial information optimal control of mean-field forward-backward stochastic system driven by Teugels martingales with applications. Neurocomputing 200, 11-2 (2016)
[9] Hao, T., Meng, Q.: Singular optimal control problems with recursive utilities of mean-field type. Asian J. Control 23(3), 1524-1535 (2021)
[10] Hu, M.: Stochastic global maximum principle for optimization with recursive utilities. Prob. Uncertain. Quant. Risk 2(1), 1 (2017)
[11] Huang, J., Shi, J.: Maximum principle for optimal control of fully coupled forward-backward stochastic differential delayed equations. ESAIM Control Optim. Calc. Var. 18(4), 1073-1096 (2012)
[12] Huang, J., Wang, B.C., Yong, J.: Social optima in mean field linear-quadratic-Gaussian control with volatility uncertainty. SIAM J. Control. Optim. 59(2), 825-856 (2021)
[13] Kac, M.: Foundations of kinetic theory. In: Proceedings of the 3rd Berkeley Symposium on Mathematical Statistics and Probability, vol. 3, pp. 171-197 (1956)
[14] Kizilkale, A.C., Salhab, R., Malham, R.P.: An integral control formulation of mean field game based large scale coordination of loads in smart grids. Automatica 100, 312-322 (2019)
[15] Li, J., Wei, Q.: Optimal control problems of fully coupled FBSDEs and viscosity solutions of Hamilton-Jacobi-Bellman dquations. SIAM J. Control. Optim. 52(3), 1622-1662 (2014)
[16] Li, R., Liu, B.: Necessary and sufficient near-optimal conditions for mean-field singular stochastic controls. Asian J. Control 17(4), 1209-1221 (2015)
[17] Li, X., Shi, J., Yong, J.: Mean-field linear-quadratic stochastic differential games in an infinite horizon. ESAIM Control Optim. Calc. Var. 27, 81 (2021)
[18] Li, X., Sun, J., Xiong, J.: Linear quadratic optimal control problems for mean-field backward stochastic differential equations. Appl. Math. Optim. 80(1), 223-250 (2019)
[19] Ma, H., Liu, B.: Linear-quadratic optimal control problem for partially observed forward-backward stochastic differential equations of mean-field type. Asian J. Control 18(6), 2146-2157 (2016)
[20] Ma, H., Liu, B.: Optimal control problem for risk-sensitive mean-field stochastic delay differential equation with partial information. Asian J. Control 19(6), 2097-2115 (2017)
[21] Ma, J., Wu, Z., Zhang, D., Zhang, J.: On well-posedness of forward-backward SDEs—a unified approach. Ann. Appl. Probab. 25(4), 2168-2214 (2015)
[22] Ma, L., Zhang, T., Zhang, W.:

\begin{document}$ H_{\infty } $\end{document}

control for continuous-time mean-field stochastic systems. Asian J. Control 18(5), 1630-1640 (2016)
[23] McKean, H.P., Jr.: A class of Markov processes associated with nonlinear parabolic equations. Proc. Natl. Acad. Sci. USA 56(6), 1907 (1966)
[24] Meng, Q., Shi, Q., Tang, M.: Maximum principle of forward-backward stochastic differential system of mean-field type with observation noise. (2017). arXiv:1708.05663
[25] Meyer-Brandis, T., Øksendal, B., Zhou, X.Y.: A mean-field stochastic maximum principle via Malliavin calculus. Stoch. Int. J. Prob. Stoch. Process. 84(5/6), 643-666 (2012)
[26] Mou, L., Yong, J.: A variational formula for stochastic controls and some applications. Pure Appl. Math. Q. 3(2), 539-567 (2007)
[27] Øksendal, B., Sulem, A.: Maximum principles for optimal control of forward-backward stochastic differential equations with jumps. SIAM J. Control. Optim. 48(5), 2945-2976 (2010)
[28] Peng, S., Wu, Z.: Fully coupled forward-backward stochastic differential equations and applications to optimal control. SIAM J. Control. Optim. 37(3), 825-843 (1999)
[29] Shen, Y., Meng, Q., Shi, P.: Maximum principle for mean-field jump-diffusion stochastic delay differential equations and its application to finance. Automatica 50(6), 1565-1579 (2014)
[30] Song, T., Liu, B.: Discrete-time mean-field stochastic linear-quadratic optimal control problem with finite horizon. Asian J. Control 23(2), 979-989 (2021)
[31] Tang, M., Meng, Q.: Linear-quadratic optimal control problems for mean-field stochastic differential equations with jumps. Asian J. Control 21(2), 809-823 (2019)
[32] Wang, B., Huang, M.: Mean-field production output control with sticky prices: Nash and social solutions. Automatica 100, 90-98 (2019)
[33] Wang, B.C., Zhang, H.: Indefinite linear quadratic mean field social control problems with multiplicative noise. IEEE Trans. Autom. Control 66(11), 5221-5236 (2020)
[34] Wang, B.C., Zhang, H., Zhang, J.F.: Mean field linear quadratic control: uniform stabilization and social optimality. Automatica 121, 109088 (2020)
[35] Wang, G., Wu, Z.: A maximum principle for mean-field stochastic control system with noisy observation. Automatica 137, 110135 (2022)
[36] Wang, G., Xiao, H., Xing, G.: An optimal control problem for mean-field forward-backward stochastic differential equation with noisy observation. Automatica 86, 104-109 (2017)
[37] Wang, G., Zhang, H.: Mean-field backward stochastic differential equation with non-Lipschitz coefficient. Asian J. Control 22(5), 1986-1994 (2020)
[38] Wu, Z.: A general maximum principle for optimal control of forward-backward stochastic systems. Automatica 49(5), 1473-1480 (2013)
[39] Ye, W., Yu, Z.: Exact controllability of linear mean-field stochastic systems and observability inequality for mean-field backward stochastic differential equations. Asian J. Control 24(1), 237-248 (2022)
[40] Yong, J.: A stochastic linear quadratic optimal control problem with generalized expectation. Stoch. Anal. Appl. 26(6), 1136-1160 (2008)
[41] Yong, J.: Optimality variational principle for controlled forward-backward stochastic differential equations with mixed initial-terminal conditions. SIAM J. Control. Optim. 48(6), 4119-4156 (2010)
[42] Yong, J.: Linear-quadratic optimal control problems for mean-field stochastic differential equations. SIAM J. Control. Optim. 51(4), 2809-2838 (2013)
[43] Yu, W., Wang, F., Huang, Y., Liu, H.: Social optimal mean field control problem for population growth model. Asian J. Control 22(6), 2444-2451 (2020)
[44] Zhang, H., Qi, Q., Fu, M.: Optimal stabilization control for discrete-time mean-field stochastic systems. IEEE Trans. Autom. Control 64(3), 1125-1136 (2018)
[45] Zhang, S., Xiong, J., Liu, X.: Stochastic maximum principle for partially observed forward-backward stochastic differential equations with jumps and regime switching. Sci. China Inf. Sci. 61(7), 1-13 (2018)
[46] Zhou, Q., Ren, Y., Wu, W.: On optimal mean-field control problem of mean-field forward-backward stochastic system with jumps under partial information. J. Syst. Sci. Complex. 30(4), 828-856 (2017)