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Backwardforward linearquadratic meanfield 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 
References:
[1] 
Andersson, D, Djehiche, B:A maximum principle for SDEs of meanfield type, Appl. Math. Optim 63, 341356 (2011) Google Scholar 
[2] 
Antonelli, F:Backwardforward stochastic differential equations. Ann. Appl. Probab 3, 777793 (1993) Google Scholar 
[3] 
Bardi, M:Explicit solutions of some linearquadratic mean field games. Netw. Heterog. Media 7, 243261(2012) Google Scholar 
[4] 
Bensoussan, A, Sung, K, Yam, S, Yung, S:Linearquadratic meanfield games. J. Optim. Theory Appl 169, 496529 (2016) Google Scholar 
[5] 
Bismut, J:An introductory approach to duality in optimal stochastic control. SIAM Rev 20, 6278 (1978) Google Scholar 
[6] 
Buckdahn, R, Cardaliaguet, P, Quincampoix, M:Some recent aspects of differential game theory. Dynam Games Appl 1, 74114 (2010) Google Scholar 
[7] 
Buckdahn, R, Djehiche, B, Li, J:A general stochastic maximum principle for SDEs of meanfield type.Appl. Math. Optim 64, 197216 (2011) Google Scholar 
[8] 
Buckdahn, R, Djehiche, B, Li, J, Peng, S:Meanfield backward stochastic differential equations:a limit approach. Ann. Probab 37, 15241565 (2009a) Google Scholar 
[9] 
Buckdahn, R, Li, J, Peng, S:Meanfield backward stochastic differential equations and related partial differential equations, Stoch. Process. Appl 119, 31333154 (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, 451492 (2014) Google Scholar 
[11] 
Carmona, R, Delarue, F:Probabilistic analysis of meanfield games. SIAM J. Control Optim 51, 27052734 (2013) Google Scholar 
[12] 
Cvitanić, J, Ma, J:Hedging options for a large investor and forwardbackward SDE's. Ann. Appl. Probab 6, 370398 (1996) Google Scholar 
[13] 
Duffie, D, Epstein, L:Stochastic differential utility. Econometrica 60, 353394 (1992) Google Scholar 
[14] 
El Karoui, N, Peng, S, Quenez, M:Backward stochastic differential equations in finance. Math.Finance 7, 171 (1997) Google Scholar 
[15] 
Espinosa, G, Touzi, N:Optimal investment under relative performance concerns. Math. Finance 25, 221257 (2015) Google Scholar 
[16] 
Guéant, O, Lasry, JM, Lions, PL:Mean field games and applications, ParisPrinceton lectures on mathematical finance. Springer, Berlin (2010) Google Scholar 
[17] 
Huang, M:Largepopulation LQG games involving a major player:the Nash certainty equivalence principle. SIAM J. Control Optim 48, 33183353 (2010) Google Scholar 
[18] 
Huang, M, Caines, P, Malhamé, R:Largepopulation costcoupled LQG problems with nonuniform agents:individualmass behavior and decentralized εNash equilibria. IEEE Trans. Autom. Control 52, 15601571 (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, 17361751 (2012) Google Scholar 
[20] 
Huang, M, Malhamé, R, Caines, P:Large population stochastic dynamic games:closedloop McKeanVlasov systems and the Nash certainty equivalence principle. Commun. Inf. Syst 6, 221251(2006) Google Scholar 
[21] 
Hu, Y, Peng, S:Solution of forwardbackward stochastic differential equations. Proba. Theory Rel. Fields 103, 273283 (1995) Google Scholar 
[22] 
Lasry, JM, Lions, PL:Mean field games. Japan J. Math 2, 229260 (2007) Google Scholar 
[23] 
Li, T, Zhang, J:Asymptotically optimal decentralized control for large population stochastic multiagent systems. IEEE Trans. Autom. Control 53, 16431660 (2008) Google Scholar 
[24] 
Lim, E, Zhou, XY:Linearquadratic control of backward stochastic differential equations. SIAM J.Control Optim 40, 450474 (2001) Google Scholar 
[25] 
Ma, J, Protter, P, Yong, J:Solving forwardbackward stochastic differential equations explicitlya four step scheme, Proba. Theory Rel. Fields 98, 339359 (1994) Google Scholar 
[26] 
Ma, J, Wu, Z, Zhang, D, Zhang, J:On wellposedness of forwardbackward SDEsa unified approach.Ann. Appl. Probab 25, 21682214 (2015) Google Scholar 
[27] 
Ma, J, Yong, J:ForwardBackward Stochastic Differential Equations and Their Applications. SpringerVerlag, Berlin Heidelberg (1999) Google Scholar 
[28] 
Nguyen, S, Huang, M:LinearquadraticGaussian mixed games with continuumparametrized minor players. SIAM J. Control Optim 50, 29072937 (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, 33023331 (2013) Google Scholar 
[30] 
Pardoux, E, Peng, S:Adapted solution of backward stochastic equation. Syst. Control Lett 14, 5561(1990) Google Scholar 
[31] 
Peng, S, Wu, Z:Fully coupled forwardbackward stochastic differential equations and applications to optimal control, SIAM. J. Control Optim 37, 825843 (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, 12301242 (2009) Google Scholar 
[33] 
Wu, Z:A general maximum principle for optimal control of forwardbackward stochastic systems.Automatica 49, 14731480 (2013) Google Scholar 
[34] 
Yong, J:Finding adapted solutions of forwardbackward stochastic differential equations:method of continuation. Proba. Theory Rel. Fields 107, 537572 (1997) Google Scholar 
[35] 
Yong, J:Optimality variational principle for controlled forwardbackward stochastic differential equations with mixed initialterminal conditions. SIAM J. Control Optim 48, 41194156 (2010) Google Scholar 
[36] 
Yong, J, Zhou, XY:Stochastic Controls:Hamiltonian Systems and HJB Equations. SpringerVerlag, New York (1999) Google Scholar 
[37] 
Yu, Z:Linearquadratic optimal control and nonzerosum differential game of forwardbackward stochastic system. Asian J.Control. 14, 173185 (2012) Google Scholar 
show all references
References:
[1] 
Andersson, D, Djehiche, B:A maximum principle for SDEs of meanfield type, Appl. Math. Optim 63, 341356 (2011) Google Scholar 
[2] 
Antonelli, F:Backwardforward stochastic differential equations. Ann. Appl. Probab 3, 777793 (1993) Google Scholar 
[3] 
Bardi, M:Explicit solutions of some linearquadratic mean field games. Netw. Heterog. Media 7, 243261(2012) Google Scholar 
[4] 
Bensoussan, A, Sung, K, Yam, S, Yung, S:Linearquadratic meanfield games. J. Optim. Theory Appl 169, 496529 (2016) Google Scholar 
[5] 
Bismut, J:An introductory approach to duality in optimal stochastic control. SIAM Rev 20, 6278 (1978) Google Scholar 
[6] 
Buckdahn, R, Cardaliaguet, P, Quincampoix, M:Some recent aspects of differential game theory. Dynam Games Appl 1, 74114 (2010) Google Scholar 
[7] 
Buckdahn, R, Djehiche, B, Li, J:A general stochastic maximum principle for SDEs of meanfield type.Appl. Math. Optim 64, 197216 (2011) Google Scholar 
[8] 
Buckdahn, R, Djehiche, B, Li, J, Peng, S:Meanfield backward stochastic differential equations:a limit approach. Ann. Probab 37, 15241565 (2009a) Google Scholar 
[9] 
Buckdahn, R, Li, J, Peng, S:Meanfield backward stochastic differential equations and related partial differential equations, Stoch. Process. Appl 119, 31333154 (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, 451492 (2014) Google Scholar 
[11] 
Carmona, R, Delarue, F:Probabilistic analysis of meanfield games. SIAM J. Control Optim 51, 27052734 (2013) Google Scholar 
[12] 
Cvitanić, J, Ma, J:Hedging options for a large investor and forwardbackward SDE's. Ann. Appl. Probab 6, 370398 (1996) Google Scholar 
[13] 
Duffie, D, Epstein, L:Stochastic differential utility. Econometrica 60, 353394 (1992) Google Scholar 
[14] 
El Karoui, N, Peng, S, Quenez, M:Backward stochastic differential equations in finance. Math.Finance 7, 171 (1997) Google Scholar 
[15] 
Espinosa, G, Touzi, N:Optimal investment under relative performance concerns. Math. Finance 25, 221257 (2015) Google Scholar 
[16] 
Guéant, O, Lasry, JM, Lions, PL:Mean field games and applications, ParisPrinceton lectures on mathematical finance. Springer, Berlin (2010) Google Scholar 
[17] 
Huang, M:Largepopulation LQG games involving a major player:the Nash certainty equivalence principle. SIAM J. Control Optim 48, 33183353 (2010) Google Scholar 
[18] 
Huang, M, Caines, P, Malhamé, R:Largepopulation costcoupled LQG problems with nonuniform agents:individualmass behavior and decentralized εNash equilibria. IEEE Trans. Autom. Control 52, 15601571 (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, 17361751 (2012) Google Scholar 
[20] 
Huang, M, Malhamé, R, Caines, P:Large population stochastic dynamic games:closedloop McKeanVlasov systems and the Nash certainty equivalence principle. Commun. Inf. Syst 6, 221251(2006) Google Scholar 
[21] 
Hu, Y, Peng, S:Solution of forwardbackward stochastic differential equations. Proba. Theory Rel. Fields 103, 273283 (1995) Google Scholar 
[22] 
Lasry, JM, Lions, PL:Mean field games. Japan J. Math 2, 229260 (2007) Google Scholar 
[23] 
Li, T, Zhang, J:Asymptotically optimal decentralized control for large population stochastic multiagent systems. IEEE Trans. Autom. Control 53, 16431660 (2008) Google Scholar 
[24] 
Lim, E, Zhou, XY:Linearquadratic control of backward stochastic differential equations. SIAM J.Control Optim 40, 450474 (2001) Google Scholar 
[25] 
Ma, J, Protter, P, Yong, J:Solving forwardbackward stochastic differential equations explicitlya four step scheme, Proba. Theory Rel. Fields 98, 339359 (1994) Google Scholar 
[26] 
Ma, J, Wu, Z, Zhang, D, Zhang, J:On wellposedness of forwardbackward SDEsa unified approach.Ann. Appl. Probab 25, 21682214 (2015) Google Scholar 
[27] 
Ma, J, Yong, J:ForwardBackward Stochastic Differential Equations and Their Applications. SpringerVerlag, Berlin Heidelberg (1999) Google Scholar 
[28] 
Nguyen, S, Huang, M:LinearquadraticGaussian mixed games with continuumparametrized minor players. SIAM J. Control Optim 50, 29072937 (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, 33023331 (2013) Google Scholar 
[30] 
Pardoux, E, Peng, S:Adapted solution of backward stochastic equation. Syst. Control Lett 14, 5561(1990) Google Scholar 
[31] 
Peng, S, Wu, Z:Fully coupled forwardbackward stochastic differential equations and applications to optimal control, SIAM. J. Control Optim 37, 825843 (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, 12301242 (2009) Google Scholar 
[33] 
Wu, Z:A general maximum principle for optimal control of forwardbackward stochastic systems.Automatica 49, 14731480 (2013) Google Scholar 
[34] 
Yong, J:Finding adapted solutions of forwardbackward stochastic differential equations:method of continuation. Proba. Theory Rel. Fields 107, 537572 (1997) Google Scholar 
[35] 
Yong, J:Optimality variational principle for controlled forwardbackward stochastic differential equations with mixed initialterminal conditions. SIAM J. Control Optim 48, 41194156 (2010) Google Scholar 
[36] 
Yong, J, Zhou, XY:Stochastic Controls:Hamiltonian Systems and HJB Equations. SpringerVerlag, New York (1999) Google Scholar 
[37] 
Yu, Z:Linearquadratic optimal control and nonzerosum differential game of forwardbackward stochastic system. Asian J.Control. 14, 173185 (2012) Google Scholar 
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