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 and Related Fields, doi: 10.3934/mcrf.2021047
References:
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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.

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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.

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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.

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D. Duffie and H. R. Richardson, Mean-variance hedging in continuous time, The Annals of Applied Probability, 1 (1991), 1-15. 

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G. GneccoM. 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.

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T. Groves, Incentives in teams, Econometrica, 41 (1973), 617-631.  doi: 10.2307/1914085.

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T. HamidouQ. Zhu and T. Başar, Risk-sensitive mean-field games, IEEE Transactions on Automatic Control, 59 (2014), 835-850.  doi: 10.1109/TAC.2013.2289711.

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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.

[12]

Y. HuJ. 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.

[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.

[14]

J. HuangG. 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.

[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.

[16]

M. HuangP. 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.

[17]

M. HuangP. 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.

[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.

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[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.

[22]

J. Marschak, Elements for a theory of teams, Management Science, 1 (1955), 127-137.  doi: 10.1287/mnsc.1.2.127.

[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.

[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.

[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.

[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.

[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.

[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.

[29]

R. Radner, Team decision problems, The Annals of Mathematical Statistics, 33 (1962), 857-881.  doi: 10.1214/aoms/1177704455.

[30]

J. SunX. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[5]

A. BensoussanK. C. J. SungS. 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.

[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.

[7]

D. Duffie and H. R. Richardson, Mean-variance hedging in continuous time, The Annals of Applied Probability, 1 (1991), 1-15. 

[8]

G. GneccoM. 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.

[9]

T. Groves, Incentives in teams, Econometrica, 41 (1973), 617-631.  doi: 10.2307/1914085.

[10]

T. HamidouQ. Zhu and T. Başar, Risk-sensitive mean-field games, IEEE Transactions on Automatic Control, 59 (2014), 835-850.  doi: 10.1109/TAC.2013.2289711.

[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.

[12]

Y. HuJ. 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.

[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.

[14]

J. HuangG. 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.

[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.

[16]

M. HuangP. 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.

[17]

M. HuangP. 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.

[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.

[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.

[20]

R. LauR. 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.

[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.

[22]

J. Marschak, Elements for a theory of teams, Management Science, 1 (1955), 127-137.  doi: 10.1287/mnsc.1.2.127.

[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.

[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.

[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.

[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.

[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.

[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.

[29]

R. Radner, Team decision problems, The Annals of Mathematical Statistics, 33 (1962), 857-881.  doi: 10.1214/aoms/1177704455.

[30]

J. SunX. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

Figure 1.  The trajectories of realized state-average $ \tilde{x}^{(N)} $ and mean-field term $ \hat{x} $
Figure 2.  The trajectories of the components in equation (41), when $ C,F\neq 0 $
Figure 3.  The trajectories of the components in equation (41), when C = F = 0
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