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Time-inconsistent stochastic optimal control problems: A backward stochastic partial differential equations approach
doi: 10.3934/mcrf.2020025

## Linear-quadratic-Gaussian mean-field-game with partial observation and common noise

 1 International Center for Decision and Risk Analysis Jindal School of Management, The University of Texas at Dallas and School of Data Sciences, City University of Hong Kong 2 Zhongtai Securities Institute for Financial Studies, Shandong University, Jinan, Shandong 250100, China 3 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong

Received  January 2019 Revised  February 2020 Published  May 2020

This paper considers a class of linear-quadratic-Gaussian (LQG) mean-field games (MFGs) with partial observation structure for individual agents. Unlike other literature, there are some special features in our formulation. First, the individual state is driven by some common-noise due to the external factor and the state-average thus becomes a random process instead of a deterministic quantity. Second, the sensor function of individual observation depends on state-average thus the agents are coupled in triple manner: not only in their states and cost functionals, but also through their observation mechanism. The decentralized strategies for individual agents are derived by the Kalman filtering and separation principle. The consistency condition is obtained which is equivalent to the wellposedness of some forward-backward stochastic differential equation (FBSDE) driven by common noise. Finally, the related $\epsilon$-Nash equilibrium property is verified.

Citation: Alain Bensoussan, Xinwei Feng, Jianhui Huang. Linear-quadratic-Gaussian mean-field-game with partial observation and common noise. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2020025
##### References:
 [1] M. Bardi, Explicit solutions of some linear-quadratic mean field games, Netw. Heterog. Media, 7 (2012), 243-261.  doi: 10.3934/nhm.2012.7.243.  Google Scholar [2] A. Bensoussan, Stochastic Control of Partially Observable Systems, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511526503.  Google Scholar [3] A. Bensoussan, J. Frehse and P. Yam, Mean Field Games and Mean Field Type Control Theory, SpringerBriefs in Mathematics, Springer, New York, 2013. doi: 10.1007/978-1-4614-8508-7.  Google Scholar [4] A. Bensoussan, K. C. J. Sung, S. C. P. Yam and S. P. Yung, Linear-quadratic mean field games, J. Optim. Theory Appl., 169 (2016), 496-529.  doi: 10.1007/s10957-015-0819-4.  Google Scholar [5] R. Carmona and F. Delarue, Probabilistic analysis of mean-field games, SIAM J. Control Optim., 51 (2013), 2705-2734.  doi: 10.1137/120883499.  Google Scholar [6] R. Carmona and F. Delarue, Probabilistic theory of mean field games with applications. Ⅱ. Mean field games with common noise and master equations, in Probability Theory and Stochastic Modelling, 84, Springer, Cham, 2018.  Google Scholar [7] R. Carmona, J.-P. Fouque and L.-H. Sun, Mean field games and systemic risk, Commun. Math. Sci., 13 (2015), 911-933.  doi: 10.4310/CMS.2015.v13.n4.a4.  Google Scholar [8] C. Dogbé, Modeling crowd dynamics by the mean-field limit approach, Math. Comput. Modelling, 52 (2010), 1506-1520.  doi: 10.1016/j.mcm.2010.06.012.  Google Scholar [9] G. M. Erickson, Differential game methods of advertising competition, European Journal Operational Research, 83 (1995), 431-438.   Google Scholar [10] W. Fleming and W. Rishel, Deterministic and Stochastic Control of Partially Observable Systems, Springer-Verlag, 1975. Google Scholar [11] O. Guéant, J.-M. Lasry and P.-L. Lions, Mean field games and applications, Paris-Princeton Lectures on Mathematical Finance 2010, Lecture Notes in Mathematics, 2003, Springer, Berlin, 2011,205–266. doi: 10.1007/978-3-642-14660-2_3.  Google Scholar [12] A. Haurie and P. Marcotte, On the relationship between Nash-Cournot and Wardrop equilibria, Networks, 15 (1985), 295-308.  doi: 10.1002/net.3230150303.  Google Scholar [13] G.-D. Hu, Symplectic Runge-Kutta methods for the linear quadratic regulator problem, Int. J. Math. Anal. (Ruse), 1 (2007), 293-304.   Google Scholar [14] J. Huang, Y. Hu and T. Nie, Linear-quadratic-Gaussian mixed mean-field games with heterogeneous input constraints, SIAM J. Control Optim., 56 (2018), 2835-2877.  doi: 10.1137/17M1151420.  Google Scholar [15] J. Huang and S. Wang, Dynamic optimization of large-population systems with partial information, J. Optim. Theory Appl., 168 (2016), 231-245.  doi: 10.1007/s10957-015-0740-x.  Google Scholar [16] M. Huang, Large-population LQG games involving a major player: The Nash certainty equivalence principle, SIAM J. Control Optim., 48 (2010), 3318-3353.  doi: 10.1137/080735370.  Google Scholar [17] M. Huang, P. E. Caines and R. P. Malhamé, Uplink power adjustment in wireless communication systems: A stochastic control analysis, IEEE Trans. Automat. Control, 49 (2004), 1693-1708.  doi: 10.1109/TAC.2004.835388.  Google Scholar [18] M. Huang, P. E. Caines and R. P. Malhamé, Distributed multi-agent decision-making with partial observations: Asymptotic Nash equilibria, Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems (MTNS), (2006), 2725–2730. Google Scholar [19] M. Huang, P. E. Caines and R. P. Malhamé, Large-population cost-coupled LQG problems with non-uniform agents: Individual-mass behavior and decentralized $\epsilon$-Nash equilibria, IEEE Trans. Automat. Control, 52 (2007), 1560-1571.  doi: 10.1109/TAC.2007.904450.  Google Scholar [20] M. Huang, R. P. Malhamé and P. E. Caines, Large population stochastic dynamic games: Closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle, Commun. Inf. Syst., 6 (2006), 221-251.  doi: 10.4310/CIS.2006.v6.n3.a5.  Google Scholar [21] G. Kallianpur, Stochastic filtering theory, in Applications of Mathematics, 13, Springer-Verlag, New York-Berlin, 1980.  Google Scholar [22] A. C. Kizilkale and R. P. Malhamé, Collective target tracking mean field control for Markovian jump-driven models of electric water heating loads, in Control of Complex Systems: Theory and Applications, 2016,559–584. Google Scholar [23] P. E. Kloeden and E. Platen, Numerical solution of sochastic differential equations, in Applications of Mathematics (New York), 23, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-12616-5.  Google Scholar [24] V. E. Lambson, Self-enforcing collusion in large dynamic markets, J. Econom. Theory, 34 (1984), 282-291.  doi: 10.1016/0022-0531(84)90145-5.  Google Scholar [25] J.-M. Lasry and P.-L. Lions, Mean field games, Jpn. J. Math., 2 (2007), 229-260.  doi: 10.1007/s11537-007-0657-8.  Google Scholar [26] J. Ma and J. Yong, Forward-backward stochastic differential equations and their applications, in Lecture Notes in Mathematics, 1702, Springer-Verlag, Berlin, 1999.  Google Scholar [27] Z. Ma, D. Callaway and I. Hiskens, Decentralized charging control of large populations of plug-in electric vehicles, IEEE Transactions on Control Systems Technology, 21 (2013), 67-78.   Google Scholar [28] S. L. Nguyen and M. Huang, Linear-quadratic-Gaussian mixed games with continuum-parameterized minor players, SIAM J. Control Optim., 50 (2012), 2907-2937.  doi: 10.1137/110841217.  Google Scholar [29] M. Nourian, P. E. Caines, R. P. Malhamé and M. Huang, Nash, social and centralized solutions to consensus problems via mean field control theory, IEEE Trans. Automat. Control, 58 (2013), 639-653.  doi: 10.1109/TAC.2012.2215399.  Google Scholar [30] B. Øksendal, Stochastic Differential Equations. An Introduction with Applications, Universitext, $6^th$ edition, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-642-14394-6.  Google Scholar [31] N. Şen and P. E. Caines, Nonlinear filtering theory for McKean–Vlasov type stochastic differential equations, SIAM J. Control Optim., 54 (2016), 153-174.  doi: 10.1137/15M1013304.  Google Scholar [32] H. Tembine, Q. Zhu and T. Başar, Risk-sensitive mean-field games, IEEE Trans. Automat. Control, 59 (2014), 835-850.  doi: 10.1109/TAC.2013.2289711.  Google Scholar [33] G. Wang and Z. Wu, Kalman-Bucy filtering equations of forward and backward stochastic systems and applications to recursive optimal control problems, J. Math. Anal. Appl., 342 (2008), 1280-1296.  doi: 10.1016/j.jmaa.2007.12.072.  Google Scholar [34] G. Wang, Z. Wu and J. Xiong, A linear-quadratic optimal control problem of forward-backward stochastic differential equations with partial information, IEEE Trans. Automat. Control, 60 (2015), 2904-2916.  doi: 10.1109/TAC.2015.2411871.  Google Scholar [35] Y. Weintraub, L. Benkard and B. Van Roy, Markov perfect industry dynamics with many firms, Econometrica, 76 (2008), 1375-1411.  doi: 10.3982/ECTA6158.  Google Scholar [36] W. M. Wonham., On the separation theorem of stochastic control., SIAM J. Control Optim., 6 (1968), 312–326. doi: 10.1137/0306023.  Google Scholar [37] H. Yin, P. G. Mehta, S. P. Meyn and U. V. Shanbhag, Synchronization of coupled oscillators is a game, IEEE Trans. Automat. Control, 57 (2012), 920-935.  doi: 10.1109/TAC.2011.2168082.  Google Scholar [38] J. Yong and X. Y. Zhou, Stochastic controls. Hamiltonian systems and HJB equations, in Applications of Mathematics (New York), 43, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3.  Google Scholar

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##### References:
 [1] M. Bardi, Explicit solutions of some linear-quadratic mean field games, Netw. Heterog. Media, 7 (2012), 243-261.  doi: 10.3934/nhm.2012.7.243.  Google Scholar [2] A. Bensoussan, Stochastic Control of Partially Observable Systems, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511526503.  Google Scholar [3] A. Bensoussan, J. Frehse and P. Yam, Mean Field Games and Mean Field Type Control Theory, SpringerBriefs in Mathematics, Springer, New York, 2013. doi: 10.1007/978-1-4614-8508-7.  Google Scholar [4] A. Bensoussan, K. C. J. Sung, S. C. P. Yam and S. P. Yung, Linear-quadratic mean field games, J. Optim. Theory Appl., 169 (2016), 496-529.  doi: 10.1007/s10957-015-0819-4.  Google Scholar [5] R. Carmona and F. Delarue, Probabilistic analysis of mean-field games, SIAM J. Control Optim., 51 (2013), 2705-2734.  doi: 10.1137/120883499.  Google Scholar [6] R. Carmona and F. Delarue, Probabilistic theory of mean field games with applications. Ⅱ. Mean field games with common noise and master equations, in Probability Theory and Stochastic Modelling, 84, Springer, Cham, 2018.  Google Scholar [7] R. Carmona, J.-P. Fouque and L.-H. Sun, Mean field games and systemic risk, Commun. Math. Sci., 13 (2015), 911-933.  doi: 10.4310/CMS.2015.v13.n4.a4.  Google Scholar [8] C. Dogbé, Modeling crowd dynamics by the mean-field limit approach, Math. Comput. Modelling, 52 (2010), 1506-1520.  doi: 10.1016/j.mcm.2010.06.012.  Google Scholar [9] G. M. Erickson, Differential game methods of advertising competition, European Journal Operational Research, 83 (1995), 431-438.   Google Scholar [10] W. Fleming and W. Rishel, Deterministic and Stochastic Control of Partially Observable Systems, Springer-Verlag, 1975. Google Scholar [11] O. Guéant, J.-M. Lasry and P.-L. Lions, Mean field games and applications, Paris-Princeton Lectures on Mathematical Finance 2010, Lecture Notes in Mathematics, 2003, Springer, Berlin, 2011,205–266. doi: 10.1007/978-3-642-14660-2_3.  Google Scholar [12] A. Haurie and P. Marcotte, On the relationship between Nash-Cournot and Wardrop equilibria, Networks, 15 (1985), 295-308.  doi: 10.1002/net.3230150303.  Google Scholar [13] G.-D. Hu, Symplectic Runge-Kutta methods for the linear quadratic regulator problem, Int. J. Math. Anal. (Ruse), 1 (2007), 293-304.   Google Scholar [14] J. Huang, Y. Hu and T. Nie, Linear-quadratic-Gaussian mixed mean-field games with heterogeneous input constraints, SIAM J. Control Optim., 56 (2018), 2835-2877.  doi: 10.1137/17M1151420.  Google Scholar [15] J. Huang and S. Wang, Dynamic optimization of large-population systems with partial information, J. Optim. Theory Appl., 168 (2016), 231-245.  doi: 10.1007/s10957-015-0740-x.  Google Scholar [16] M. Huang, Large-population LQG games involving a major player: The Nash certainty equivalence principle, SIAM J. Control Optim., 48 (2010), 3318-3353.  doi: 10.1137/080735370.  Google Scholar [17] M. Huang, P. E. Caines and R. P. Malhamé, Uplink power adjustment in wireless communication systems: A stochastic control analysis, IEEE Trans. Automat. Control, 49 (2004), 1693-1708.  doi: 10.1109/TAC.2004.835388.  Google Scholar [18] M. Huang, P. E. Caines and R. P. Malhamé, Distributed multi-agent decision-making with partial observations: Asymptotic Nash equilibria, Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems (MTNS), (2006), 2725–2730. Google Scholar [19] M. Huang, P. E. Caines and R. P. Malhamé, Large-population cost-coupled LQG problems with non-uniform agents: Individual-mass behavior and decentralized $\epsilon$-Nash equilibria, IEEE Trans. Automat. Control, 52 (2007), 1560-1571.  doi: 10.1109/TAC.2007.904450.  Google Scholar [20] M. Huang, R. P. Malhamé and P. E. Caines, Large population stochastic dynamic games: Closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle, Commun. Inf. Syst., 6 (2006), 221-251.  doi: 10.4310/CIS.2006.v6.n3.a5.  Google Scholar [21] G. Kallianpur, Stochastic filtering theory, in Applications of Mathematics, 13, Springer-Verlag, New York-Berlin, 1980.  Google Scholar [22] A. C. Kizilkale and R. P. Malhamé, Collective target tracking mean field control for Markovian jump-driven models of electric water heating loads, in Control of Complex Systems: Theory and Applications, 2016,559–584. Google Scholar [23] P. E. Kloeden and E. Platen, Numerical solution of sochastic differential equations, in Applications of Mathematics (New York), 23, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-12616-5.  Google Scholar [24] V. E. Lambson, Self-enforcing collusion in large dynamic markets, J. Econom. Theory, 34 (1984), 282-291.  doi: 10.1016/0022-0531(84)90145-5.  Google Scholar [25] J.-M. Lasry and P.-L. Lions, Mean field games, Jpn. J. Math., 2 (2007), 229-260.  doi: 10.1007/s11537-007-0657-8.  Google Scholar [26] J. Ma and J. Yong, Forward-backward stochastic differential equations and their applications, in Lecture Notes in Mathematics, 1702, Springer-Verlag, Berlin, 1999.  Google Scholar [27] Z. Ma, D. Callaway and I. Hiskens, Decentralized charging control of large populations of plug-in electric vehicles, IEEE Transactions on Control Systems Technology, 21 (2013), 67-78.   Google Scholar [28] S. L. Nguyen and M. Huang, Linear-quadratic-Gaussian mixed games with continuum-parameterized minor players, SIAM J. Control Optim., 50 (2012), 2907-2937.  doi: 10.1137/110841217.  Google Scholar [29] M. Nourian, P. E. Caines, R. P. Malhamé and M. Huang, Nash, social and centralized solutions to consensus problems via mean field control theory, IEEE Trans. Automat. Control, 58 (2013), 639-653.  doi: 10.1109/TAC.2012.2215399.  Google Scholar [30] B. Øksendal, Stochastic Differential Equations. An Introduction with Applications, Universitext, $6^th$ edition, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-642-14394-6.  Google Scholar [31] N. Şen and P. E. Caines, Nonlinear filtering theory for McKean–Vlasov type stochastic differential equations, SIAM J. Control Optim., 54 (2016), 153-174.  doi: 10.1137/15M1013304.  Google Scholar [32] H. Tembine, Q. Zhu and T. Başar, Risk-sensitive mean-field games, IEEE Trans. Automat. Control, 59 (2014), 835-850.  doi: 10.1109/TAC.2013.2289711.  Google Scholar [33] G. Wang and Z. Wu, Kalman-Bucy filtering equations of forward and backward stochastic systems and applications to recursive optimal control problems, J. Math. Anal. Appl., 342 (2008), 1280-1296.  doi: 10.1016/j.jmaa.2007.12.072.  Google Scholar [34] G. Wang, Z. Wu and J. Xiong, A linear-quadratic optimal control problem of forward-backward stochastic differential equations with partial information, IEEE Trans. Automat. Control, 60 (2015), 2904-2916.  doi: 10.1109/TAC.2015.2411871.  Google Scholar [35] Y. Weintraub, L. Benkard and B. Van Roy, Markov perfect industry dynamics with many firms, Econometrica, 76 (2008), 1375-1411.  doi: 10.3982/ECTA6158.  Google Scholar [36] W. M. Wonham., On the separation theorem of stochastic control., SIAM J. Control Optim., 6 (1968), 312–326. doi: 10.1137/0306023.  Google Scholar [37] H. Yin, P. G. Mehta, S. P. Meyn and U. V. Shanbhag, Synchronization of coupled oscillators is a game, IEEE Trans. Automat. Control, 57 (2012), 920-935.  doi: 10.1109/TAC.2011.2168082.  Google Scholar [38] J. Yong and X. Y. Zhou, Stochastic controls. Hamiltonian systems and HJB equations, in Applications of Mathematics (New York), 43, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3.  Google Scholar
Trajectories of the type-1 agents' states when N = 500
Trajectories of the type-2 agents' states when N = 500
Trajectories of the type-1 agents state average and the mean field term
Trajectories of the type-2 agents state average and the mean field term
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