January  2016, 1: 8 doi: 10.1186/s41546-016-0009-9

Backward-forward linear-quadratic mean-field games with major and minor agents

1 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong kong, China;

2 School of Mathematics, Shandong University, 250100 Jinan, China

Received  April 04, 2016 Revised  September 12, 2016

Fund Project: support partly by RGC Grant 502412, 15300514, G-YL04. Z. Wu acknowledges the Natural Science Foundation of China (61573217), 111 project (B12023), the National High-level personnel of special support program and the Chang Jiang Scholar Program of Chinese Education Ministry.

This paper studies the backward-forward linear-quadratic-Gaussian (LQG) games with major and minor agents (players). The state of major agent follows a linear backward stochastic differential equation (BSDE) and the states of minor agents are governed by linear forward stochastic differential equations (SDEs). The major agent is dominating as its state enters those of minor agents. On the other hand, all minor agents are individually negligible but their state-average affects the cost functional of major agent. The mean-field game in such backward-major and forward-minor setup is formulated to analyze the decentralized strategies. We first derive the consistency condition via an auxiliary mean-field SDEs and a 3×2 mixed backward-forward stochastic differential equation (BFSDE) system. Next, we discuss the wellposedness of such BFSDE system by virtue of the monotonicity method. Consequently, we obtain the decentralized strategies for major and minor agents which are proved to satisfy the -Nash equilibrium property.
Citation: Jianhui Huang, Shujun Wang, Zhen Wu. Backward-forward linear-quadratic mean-field games with major and minor agents. Probability, Uncertainty and Quantitative Risk, doi: 10.1186/s41546-016-0009-9
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Antonelli, F:Backward-forward stochastic differential equations. Ann. Appl. Probab 3, 777-793 (1993),

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Bensoussan, A, Sung, K, Yam, S, Yung, S:Linear-quadratic mean-field games. J. Optim. Theory Appl 169, 496-529 (2016),

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Buckdahn, R, Cardaliaguet, P, Quincampoix, M:Some recent aspects of differential game theory. Dynam Games Appl 1, 74-114 (2010),

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Buckdahn, R, Djehiche, B, Li, J:A general stochastic maximum principle for SDEs of mean-field type.Appl. Math. Optim 64, 197-216 (2011),

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Buckdahn, R, Djehiche, B, Li, J, Peng, S:Mean-field backward stochastic differential equations:a limit approach. Ann. Probab 37, 1524-1565 (2009a),

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Buckdahn, R, Li, J, Peng, S:Mean-field backward stochastic differential equations and related partial differential equations, Stoch. Process. Appl 119, 3133-3154 (2009b),

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Buckdahn, R, Li, J, Peng, S:Nonlinear stochastic differential games involving a major player and a large number of collectively acting minor agents. SIAM J. Control Optim 52, 451-492 (2014),

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Carmona, R, Delarue, F:Probabilistic analysis of mean-field games. SIAM J. Control Optim 51, 2705-2734 (2013),

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Cvitanić, J, Ma, J:Hedging options for a large investor and forward-backward SDE's. Ann. Appl. Probab 6, 370-398 (1996),

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Duffie, D, Epstein, L:Stochastic differential utility. Econometrica 60, 353-394 (1992),

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El Karoui, N, Peng, S, Quenez, M:Backward stochastic differential equations in finance. Math.Finance 7, 1-71 (1997),

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Espinosa, G, Touzi, N:Optimal investment under relative performance concerns. Math. Finance 25, 221-257 (2015),

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Guéant, O, Lasry, J-M, Lions, P-L:Mean field games and applications, Paris-Princeton lectures on mathematical finance. Springer, Berlin (2010),

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Huang, M:Large-population LQG games involving a major player:the Nash certainty equivalence principle. SIAM J. Control Optim 48, 3318-3353 (2010),

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Huang, M, Caines, P, Malhamé, R:Large-population cost-coupled LQG problems with non-uniform agents:individual-mass behavior and decentralized ε-Nash equilibria. IEEE Trans. Autom. Control 52, 1560-1571 (2007),

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Huang, M, Caines, P, Malhamé, R:Social optima in mean field LQG control:centralized and decentralized strategies. IEEE Trans. Autom. Control 57, 1736-1751 (2012),

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Huang, M, Malhamé, R, Caines, P:Large population stochastic dynamic games:closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle. Commun. Inf. Syst 6, 221-251(2006),

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Hu, Y, Peng, S:Solution of forwardbackward stochastic differential equations. Proba. Theory Rel. Fields 103, 273-283 (1995),

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Lasry, J-M, Lions, P-L:Mean field games. Japan J. Math 2, 229-260 (2007),

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Li, T, Zhang, J:Asymptotically optimal decentralized control for large population stochastic multiagent systems. IEEE Trans. Autom. Control 53, 1643-1660 (2008),

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Lim, E, Zhou, XY:Linear-quadratic control of backward stochastic differential equations. SIAM J.Control Optim 40, 450-474 (2001),

[25]

Ma, J, Protter, P, Yong, J:Solving forward-backward stochastic differential equations explicitly-a four step scheme, Proba. Theory Rel. Fields 98, 339-359 (1994),

[26]

Ma, J, Wu, Z, Zhang, D, Zhang, J:On well-posedness of forward-backward SDEs-a unified approach.Ann. Appl. Probab 25, 2168-2214 (2015),

[27]

Ma, J, Yong, J:Forward-Backward Stochastic Differential Equations and Their Applications. SpringerVerlag, Berlin Heidelberg (1999),

[28]

Nguyen, S, Huang, M:Linear-quadratic-Gaussian mixed games with continuum-parametrized minor players. SIAM J. Control Optim 50, 2907-2937 (2012),

[29]

Nourian, M, Caines, P:∊-Nash mean field game theory for nonlinear stochastic dynamical systems with major and minor agents. SIAM J. Control Optim 51, 3302-3331 (2013),

[30]

Pardoux, E, Peng, S:Adapted solution of backward stochastic equation. Syst. Control Lett 14, 55-61(1990),

[31]

Peng, S, Wu, Z:Fully coupled forward-backward stochastic differential equations and applications to optimal control, SIAM. J. Control Optim 37, 825-843 (1999),

[32]

Wang, G, Wu, Z:The maximum principles for stochastic recursive optimal control problems under partial information. IEEE Trans. Autom. Control 54, 1230-1242 (2009),

[33]

Wu, Z:A general maximum principle for optimal control of forward-backward stochastic systems.Automatica 49, 1473-1480 (2013),

[34]

Yong, J:Finding adapted solutions of forward-backward stochastic differential equations:method of continuation. Proba. Theory Rel. Fields 107, 537-572 (1997),

[35]

Yong, J:Optimality variational principle for controlled forward-backward stochastic differential equations with mixed initial-terminal conditions. SIAM J. Control Optim 48, 4119-4156 (2010),

[36]

Yong, J, Zhou, XY:Stochastic Controls:Hamiltonian Systems and HJB Equations. Springer-Verlag, New York (1999),

[37]

Yu, Z:Linear-quadratic optimal control and nonzero-sum differential game of forward-backward stochastic system. Asian J.Control. 14, 173-185 (2012),

show all references

References:
[1]

Andersson, D, Djehiche, B:A maximum principle for SDEs of mean-field type, Appl. Math. Optim 63, 341-356 (2011),

[2]

Antonelli, F:Backward-forward stochastic differential equations. Ann. Appl. Probab 3, 777-793 (1993),

[3]

Bardi, M:Explicit solutions of some linear-quadratic mean field games. Netw. Heterog. Media 7, 243-261(2012),

[4]

Bensoussan, A, Sung, K, Yam, S, Yung, S:Linear-quadratic mean-field games. J. Optim. Theory Appl 169, 496-529 (2016),

[5]

Bismut, J:An introductory approach to duality in optimal stochastic control. SIAM Rev 20, 62-78 (1978),

[6]

Buckdahn, R, Cardaliaguet, P, Quincampoix, M:Some recent aspects of differential game theory. Dynam Games Appl 1, 74-114 (2010),

[7]

Buckdahn, R, Djehiche, B, Li, J:A general stochastic maximum principle for SDEs of mean-field type.Appl. Math. Optim 64, 197-216 (2011),

[8]

Buckdahn, R, Djehiche, B, Li, J, Peng, S:Mean-field backward stochastic differential equations:a limit approach. Ann. Probab 37, 1524-1565 (2009a),

[9]

Buckdahn, R, Li, J, Peng, S:Mean-field backward stochastic differential equations and related partial differential equations, Stoch. Process. Appl 119, 3133-3154 (2009b),

[10]

Buckdahn, R, Li, J, Peng, S:Nonlinear stochastic differential games involving a major player and a large number of collectively acting minor agents. SIAM J. Control Optim 52, 451-492 (2014),

[11]

Carmona, R, Delarue, F:Probabilistic analysis of mean-field games. SIAM J. Control Optim 51, 2705-2734 (2013),

[12]

Cvitanić, J, Ma, J:Hedging options for a large investor and forward-backward SDE's. Ann. Appl. Probab 6, 370-398 (1996),

[13]

Duffie, D, Epstein, L:Stochastic differential utility. Econometrica 60, 353-394 (1992),

[14]

El Karoui, N, Peng, S, Quenez, M:Backward stochastic differential equations in finance. Math.Finance 7, 1-71 (1997),

[15]

Espinosa, G, Touzi, N:Optimal investment under relative performance concerns. Math. Finance 25, 221-257 (2015),

[16]

Guéant, O, Lasry, J-M, Lions, P-L:Mean field games and applications, Paris-Princeton lectures on mathematical finance. Springer, Berlin (2010),

[17]

Huang, M:Large-population LQG games involving a major player:the Nash certainty equivalence principle. SIAM J. Control Optim 48, 3318-3353 (2010),

[18]

Huang, M, Caines, P, Malhamé, R:Large-population cost-coupled LQG problems with non-uniform agents:individual-mass behavior and decentralized ε-Nash equilibria. IEEE Trans. Autom. Control 52, 1560-1571 (2007),

[19]

Huang, M, Caines, P, Malhamé, R:Social optima in mean field LQG control:centralized and decentralized strategies. IEEE Trans. Autom. Control 57, 1736-1751 (2012),

[20]

Huang, M, Malhamé, R, Caines, P:Large population stochastic dynamic games:closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle. Commun. Inf. Syst 6, 221-251(2006),

[21]

Hu, Y, Peng, S:Solution of forwardbackward stochastic differential equations. Proba. Theory Rel. Fields 103, 273-283 (1995),

[22]

Lasry, J-M, Lions, P-L:Mean field games. Japan J. Math 2, 229-260 (2007),

[23]

Li, T, Zhang, J:Asymptotically optimal decentralized control for large population stochastic multiagent systems. IEEE Trans. Autom. Control 53, 1643-1660 (2008),

[24]

Lim, E, Zhou, XY:Linear-quadratic control of backward stochastic differential equations. SIAM J.Control Optim 40, 450-474 (2001),

[25]

Ma, J, Protter, P, Yong, J:Solving forward-backward stochastic differential equations explicitly-a four step scheme, Proba. Theory Rel. Fields 98, 339-359 (1994),

[26]

Ma, J, Wu, Z, Zhang, D, Zhang, J:On well-posedness of forward-backward SDEs-a unified approach.Ann. Appl. Probab 25, 2168-2214 (2015),

[27]

Ma, J, Yong, J:Forward-Backward Stochastic Differential Equations and Their Applications. SpringerVerlag, Berlin Heidelberg (1999),

[28]

Nguyen, S, Huang, M:Linear-quadratic-Gaussian mixed games with continuum-parametrized minor players. SIAM J. Control Optim 50, 2907-2937 (2012),

[29]

Nourian, M, Caines, P:∊-Nash mean field game theory for nonlinear stochastic dynamical systems with major and minor agents. SIAM J. Control Optim 51, 3302-3331 (2013),

[30]

Pardoux, E, Peng, S:Adapted solution of backward stochastic equation. Syst. Control Lett 14, 55-61(1990),

[31]

Peng, S, Wu, Z:Fully coupled forward-backward stochastic differential equations and applications to optimal control, SIAM. J. Control Optim 37, 825-843 (1999),

[32]

Wang, G, Wu, Z:The maximum principles for stochastic recursive optimal control problems under partial information. IEEE Trans. Autom. Control 54, 1230-1242 (2009),

[33]

Wu, Z:A general maximum principle for optimal control of forward-backward stochastic systems.Automatica 49, 1473-1480 (2013),

[34]

Yong, J:Finding adapted solutions of forward-backward stochastic differential equations:method of continuation. Proba. Theory Rel. Fields 107, 537-572 (1997),

[35]

Yong, J:Optimality variational principle for controlled forward-backward stochastic differential equations with mixed initial-terminal conditions. SIAM J. Control Optim 48, 4119-4156 (2010),

[36]

Yong, J, Zhou, XY:Stochastic Controls:Hamiltonian Systems and HJB Equations. Springer-Verlag, New York (1999),

[37]

Yu, Z:Linear-quadratic optimal control and nonzero-sum differential game of forward-backward stochastic system. Asian J.Control. 14, 173-185 (2012),

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