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

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

Jianhui Huang 1,, , Shujun Wang 2,, and  Zhen Wu 2,,

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 Published  December 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.
Keywords: Backward-forward stochastic differential equation (BFSDE), Consistency condition, ∊-Nash equilibrium, Large-population system, Major-minor agent, Mean-field game.
Citation: Jianhui Huang, Shujun Wang, Zhen Wu. Backward-forward linear-quadratic mean-field games with major and minor agents. Probability, Uncertainty and Quantitative Risk, 2016, 1 (0) : 8-. doi: 10.1186/s41546-016-0009-9
References:
[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[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) Google Scholar

[8]

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

[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) Google Scholar

[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) Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

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

[16]

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

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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) Google Scholar

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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) Google Scholar

[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) Google Scholar

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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) Google Scholar

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

[22]

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

[23]

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

[24]

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

[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) Google Scholar

[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) Google Scholar

[27]

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

[28]

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

[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) Google Scholar

[30]

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

[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) Google Scholar

[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) Google Scholar

[33]

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

[34]

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

[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) Google Scholar

[36]

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

[37]

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

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) Google Scholar

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[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) Google Scholar

[8]

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

[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) Google Scholar

[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) Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[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) Google Scholar

[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) Google Scholar

[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) Google Scholar

[21]

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

[22]

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

[23]

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

[24]

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

[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) Google Scholar

[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) Google Scholar

[27]

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

[28]

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

[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) Google Scholar

[30]

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

[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) Google Scholar

[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) Google Scholar

[33]

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

[34]

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

[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) Google Scholar

[36]

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

[37]

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

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