# American Institute of Mathematical Sciences

doi: 10.3934/mcrf.2021047
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## Linear-Quadratic-Gaussian mean-field controls of social optima

 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China

* Corresponding author: tinghan.xie@connect.polyu.hk

Received  February 2020 Revised  March 2021 Early access September 2021

Fund Project: The first author acknowledges the support from: RGC Grants PolyU P0008686; The second author is supported by the financial support by RGC Grants PolyU 153005/14P, 153275/16P, P0030808; The third author acknowledges the support from: RGC Grants PolyU P0031044

This paper investigates a class of unified stochastic linear-quadratic-Gaussian (LQG) social optima problems involving a large number of weakly-coupled interactive agents under a generalized setting. For each individual agent, the control and state process enters both diffusion and drift terms in its linear dynamics, and the control weight might be indefinite in cost functional. This setup is innovative and has great theoretical and realistic significance as its applications in mathematical finance (e.g., portfolio selection in mean-variation model). Using some fully-coupled variational analysis under the person-by-person optimality principle, and the mean-field approximation method, the decentralized social control is derived by a class of new type consistency condition (CC) system for typical representative agent. Such CC system is some mean-field forward-backward stochastic differential equation (MF-FBSDE) combined with embedding representation. The well-posedness of such forward-backward stochastic differential equation (FBSDE) system is carefully examined. The related social asymptotic optimality is related to the convergence of the average of a series of weakly-coupled backward stochastic differential equation (BSDE). They are verified through some Lyapunov equations.

Citation: Zhenghong Qiu, Jianhui Huang, Tinghan Xie. Linear-Quadratic-Gaussian mean-field controls of social optima. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2021047
##### References:
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Tang, Forward-backward stochastic differential equations and quasilinear parabolic pdes, Probability Theory and Related Fields, 114 (1999), 123-150.  doi: 10.1007/s004409970001.  Google Scholar [28] S. Peng and Z. Wu, Fully coupled forward-backward stochastic differential equations and applications to optimal control, SIAM Journal on Control and Optimization, 37 (1999), 825-843.  doi: 10.1137/S0363012996313549.  Google Scholar [29] R. Radner, Team decision problems, The Annals of Mathematical Statistics, 33 (1962), 857-881.  doi: 10.1214/aoms/1177704455.  Google Scholar [30] J. Sun, X. Li and J. Yong, Open-loop and closed-loop solvabilities for stochastic linear quadratic optimal control problems, SIAM Journal on Control and Optimization, 54 (2016), 2274-2308.  doi: 10.1137/15M103532X.  Google Scholar [31] J. Sun and J. Yong, Linear quadratic stochastic differential games: Open-loop and closed-loop saddle points, SIAM Journal on Control and Optimization, 52 (2014), 4082-4121.  doi: 10.1137/140953642.  Google Scholar [32] B.-C. Wang and J. Huang, Social optima in robust mean field lqg control, In 2017 11th Asian Control Conference (ASCC), IEEE, (2017), 2089–2094. doi: 10.1109/ASCC.2017.8287497.  Google Scholar [33] B.-C. Wang, J. Huang and J.-F. Zhang, Social optima in robust mean field lqg control: From finite to infinite horizon, IEEE Transactions on Automatic Control, 66 (2021), 1529–1544. arXiv preprint arXiv: 1908.01122, 2019. doi: 10.1109/TAC.2020.2996189.  Google Scholar [34] B.-C. Wang and J.-F. Zhang, Hierarchical mean field games for multiagent systems with tracking-type costs: Distributed $\varepsilon$-stackelberg equilibria, IEEE Transactions on Automatic Control, 59 (2014), 2241-2247.  doi: 10.1109/TAC.2014.2301576.  Google Scholar [35] B.-C. Wang and J.-F. Zhang, Social optima in mean field linear-quadratic-gaussian models with markov jump parameters, SIAM Journal on Control and Optimization, 55 (2017), 429-456.  doi: 10.1137/15M104178X.  Google Scholar [36] S.-H. Wang and E. Davison, On the stabilization of decentralized control systems, IEEE Transactions on Automatic Control, 18 (1973), 473-478.  doi: 10.1109/tac.1973.1100362.  Google Scholar [37] J. Xiong and X. Y. Zhou, Mean-variance portfolio selection under partial information, SIAM Journal on Control and Optimization, 46 (2007), 156-175.  doi: 10.1137/050641132.  Google Scholar [38] J. Yong, Linear forward-backward stochastic differential equations with random coefficients, Probability Theory and Related Fields, 135 (2006), 53-83.  doi: 10.1007/s00440-005-0452-5.  Google Scholar [39] J. Yong, Linear-quadratic optimal control problems for mean-field stochastic differential equations, SIAM Journal on Control and Optimization, 51 (2013), 2809-2838.  doi: 10.1137/120892477.  Google Scholar [40] J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, volume 43, Springer Science & Business Media, 1999. doi: 10.1007/978-1-4612-1466-3.  Google Scholar [41] X. Y. Zhou and D. Li, Continuous-time mean-variance portfolio selection: A stochastic lq framework, Applied Mathematics and Optimization, 42 (2000), 19-33.  doi: 10.1007/s002450010003.  Google Scholar

show all references

##### References:
 [1] M. Aoki, On feedback stabilizability of decentralized dynamic systems, Automatica, 8 (1972), 163-173.  doi: 10.1016/0005-1098(72)90064-7.  Google Scholar [2] C. T. Bauch and D. J. D. Earn, Vaccination and the theory of games, Proceedings of the National Academy of Sciences, 101 (2004), 13391-13394.  doi: 10.1073/pnas.0403823101.  Google Scholar [3] D. Bauso and R. Pesenti, Team theory and person-by-person optimization with binary decisions, SIAM Journal on Control and Optimization, 50 (2012), 3011-3028.  doi: 10.1137/090769533.  Google Scholar [4] 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 [5] A. Bensoussan, K. C. J. Sung, S. C. P. Yam and S. P. Yung, Linear-quadratic mean field games, Journal of Optimization Theory and Applications, 169 (2016), 496-529.  doi: 10.1007/s10957-015-0819-4.  Google Scholar [6] R. Carmona and F. Delarue, Probabilistic analysis of mean-field games, SIAM Journal on Control and Optimization, 51 (2013), 2705-2734.  doi: 10.1137/120883499.  Google Scholar [7] D. Duffie and H. R. Richardson, Mean-variance hedging in continuous time, The Annals of Applied Probability, 1 (1991), 1-15.   Google Scholar [8] G. Gnecco, M. Sanguineti and M. Gaggero, Suboptimal solutions to team optimization problems with stochastic information structure, SIAM Journal on Optimization, 22 (2012), 212-243.  doi: 10.1137/100803481.  Google Scholar [9] T. Groves, Incentives in teams, Econometrica, 41 (1973), 617-631.  doi: 10.2307/1914085.  Google Scholar [10] T. Hamidou, Q. Zhu and T. Başar, Risk-sensitive mean-field games, IEEE Transactions on Automatic Control, 59 (2014), 835-850.  doi: 10.1109/TAC.2013.2289711.  Google Scholar [11] Y. C. Ho and K. C. Chu, Team decision theory and information structures in optimal control problems–part i, IEEE Transactions on Automatic Control, 17 (1972), 15-22.  doi: 10.1109/TAC.1972.1099850.  Google Scholar [12] Y. Hu, J. Huang and T. Nie, Linear-quadratic-gaussian mixed mean-field games with heterogeneous input constraints, SIAM Journal on Control and Optimization, 56 (2018), 2835-2877.  doi: 10.1137/17M1151420.  Google Scholar [13] J. Huang and M. Huang, Robust mean field linear-quadratic-gaussian games with unknown l^2-disturbance, SIAM Journal on Control and Optimization, 55 (2017), 2811-2840.  doi: 10.1137/15M1014437.  Google Scholar [14] J. Huang, G. Wang and Z. Wu, Optimal premium policy of an insurance firm: Full and partial information, Insurance: Mathematics and Economics, 47 (2010), 208-215.  doi: 10.1016/j.insmatheco.2010.04.007.  Google Scholar [15] M. Huang, Large-population lqg games involving a major player: The nash certainty equivalence principle, SIAM Journal on Control and Optimization, 48 (2010), 3318-3353.  doi: 10.1137/080735370.  Google Scholar [16] M. Huang, P. E. Caines and R. P. Malhamé, Large-population cost-coupled lqg problems with nonuniform agents: Individual-mass behavior and decentralized $\varepsilon$-nash equilibria, IEEE Transactions on Automatic Control, 52 (2007), 1560-1571.  doi: 10.1109/TAC.2007.904450.  Google Scholar [17] M. Huang, P. E. Caines and R. P. Malhamé, Social optima in mean field lqg control: Centralized and decentralized strategies, IEEE Transactions on Automatic Control, 57 (2012), 1736-1751.  doi: 10.1109/TAC.2012.2183439.  Google Scholar [18] A. C. Kizilkale and R. P. Malhame, Collective target tracking mean field control for markovian jump-driven models of electric water heating loads, Proceedings of the 19th IFAC World Congress, Cape Town, South Africa, (2014), 1867–1972. doi: 10.3182/20140824-6-ZA-1003.00630.  Google Scholar [19] J.-M. Lasry and P.-L. Lions, Mean field games, Japanese Journal of Mathematics, 2 (2007), 229-260.  doi: 10.1007/s11537-007-0657-8.  Google Scholar [20] R. Lau, R. Persiano and P. Varaiya, Decentralized information and control: A network flow example, IEEE Transactions on Automatic Control, 17 (1972), 466-473.  doi: 10.1109/tac.1972.1100036.  Google Scholar [21] T. Li and J.-F. Zhang, Asymptotically optimal decentralized control for large population stochastic multiagent systems, IEEE Transactions on Automatic Control, 53 (2008), 1643-1660.  doi: 10.1109/TAC.2008.929370.  Google Scholar [22] J. Marschak, Elements for a theory of teams, Management Science, 1 (1955), 127-137.  doi: 10.1287/mnsc.1.2.127.  Google Scholar [23] R. C. Merton, Optimum consumption and portfolio rules in a continuous-time model, Journal of Economic Theory, 3 (1971), 373-413.  doi: 10.1016/0022-0531(71)90038-X.  Google Scholar [24] J. Moon and T. Başar, Linear quadratic risk-sensitive and robust mean field games, IEEE Transactions on Automatic Control, 62 (2017), 1062-1077.  doi: 10.1109/TAC.2016.2579264.  Google Scholar [25] J. Moon and T. Başar, Linear quadratic mean field stackelberg differential games, Automatica, 97 (2018), 200-213.  doi: 10.1016/j.automatica.2018.08.008.  Google Scholar [26] M. Nourian and P. E. Caines, $\epsilon$-nash mean field game theory for nonlinear stochastic dynamical systems with major and minor agents, SIAM Journal on Control and Optimization, 51 (2013), 3302-3331.  doi: 10.1137/120889496.  Google Scholar [27] E. Pardoux and S. Tang, Forward-backward stochastic differential equations and quasilinear parabolic pdes, Probability Theory and Related Fields, 114 (1999), 123-150.  doi: 10.1007/s004409970001.  Google Scholar [28] S. Peng and Z. Wu, Fully coupled forward-backward stochastic differential equations and applications to optimal control, SIAM Journal on Control and Optimization, 37 (1999), 825-843.  doi: 10.1137/S0363012996313549.  Google Scholar [29] R. Radner, Team decision problems, The Annals of Mathematical Statistics, 33 (1962), 857-881.  doi: 10.1214/aoms/1177704455.  Google Scholar [30] J. Sun, X. Li and J. Yong, Open-loop and closed-loop solvabilities for stochastic linear quadratic optimal control problems, SIAM Journal on Control and Optimization, 54 (2016), 2274-2308.  doi: 10.1137/15M103532X.  Google Scholar [31] J. Sun and J. Yong, Linear quadratic stochastic differential games: Open-loop and closed-loop saddle points, SIAM Journal on Control and Optimization, 52 (2014), 4082-4121.  doi: 10.1137/140953642.  Google Scholar [32] B.-C. Wang and J. Huang, Social optima in robust mean field lqg control, In 2017 11th Asian Control Conference (ASCC), IEEE, (2017), 2089–2094. doi: 10.1109/ASCC.2017.8287497.  Google Scholar [33] B.-C. Wang, J. Huang and J.-F. Zhang, Social optima in robust mean field lqg control: From finite to infinite horizon, IEEE Transactions on Automatic Control, 66 (2021), 1529–1544. arXiv preprint arXiv: 1908.01122, 2019. doi: 10.1109/TAC.2020.2996189.  Google Scholar [34] B.-C. Wang and J.-F. Zhang, Hierarchical mean field games for multiagent systems with tracking-type costs: Distributed $\varepsilon$-stackelberg equilibria, IEEE Transactions on Automatic Control, 59 (2014), 2241-2247.  doi: 10.1109/TAC.2014.2301576.  Google Scholar [35] B.-C. Wang and J.-F. Zhang, Social optima in mean field linear-quadratic-gaussian models with markov jump parameters, SIAM Journal on Control and Optimization, 55 (2017), 429-456.  doi: 10.1137/15M104178X.  Google Scholar [36] S.-H. Wang and E. Davison, On the stabilization of decentralized control systems, IEEE Transactions on Automatic Control, 18 (1973), 473-478.  doi: 10.1109/tac.1973.1100362.  Google Scholar [37] J. Xiong and X. Y. Zhou, Mean-variance portfolio selection under partial information, SIAM Journal on Control and Optimization, 46 (2007), 156-175.  doi: 10.1137/050641132.  Google Scholar [38] J. Yong, Linear forward-backward stochastic differential equations with random coefficients, Probability Theory and Related Fields, 135 (2006), 53-83.  doi: 10.1007/s00440-005-0452-5.  Google Scholar [39] J. Yong, Linear-quadratic optimal control problems for mean-field stochastic differential equations, SIAM Journal on Control and Optimization, 51 (2013), 2809-2838.  doi: 10.1137/120892477.  Google Scholar [40] J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, volume 43, Springer Science & Business Media, 1999. doi: 10.1007/978-1-4612-1466-3.  Google Scholar [41] X. Y. Zhou and D. Li, Continuous-time mean-variance portfolio selection: A stochastic lq framework, Applied Mathematics and Optimization, 42 (2000), 19-33.  doi: 10.1007/s002450010003.  Google Scholar
The trajectories of realized state-average $\tilde{x}^{(N)}$ and mean-field term $\hat{x}$
The trajectories of the components in equation (41), when $C,F\neq 0$
The trajectories of the components in equation (41), when C = F = 0

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