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Explicit solutions of some linearquadratic mean field games
1.  Dipartimento di Matematica, Università di Padova, via Trieste, 63; I35121 Padova, Italy 
References:
[1] 
Y. Achdou, F. Camilli and I. CapuzzoDolcetta, Mean field games: Numerical methods for the planning problem, SIAM J. Control Opt., 50 (2012), 77109. doi: 10.1137/100790069. 
[2] 
Y. Achdou and I. CapuzzoDolcetta, Mean field games: Numerical methods, SIAM J. Numer. Anal., 48 (2010), 11361162. doi: 10.1137/090758477. 
[3] 
O. Alvarez and M. Bardi, Ergodic problems in differential games, in "Advances in Dynamic Game Theory," Ann. Internat. Soc. Dynam. Games, 9, Birkhäuser Boston, Boston, MA, (2007), 131152. 
[4] 
O. Alvarez and M. Bardi, Ergodicity, stabilization, and singular perturbations for BellmanIsaacs equations, Mem. Amer. Math. Soc., 204 (2010), vi+77 pp. 
[5] 
R. J. Aumann, Markets with a continuum of traders, Econometrica, 32 (1964), 3950. doi: 10.2307/1913732. 
[6] 
M. Bardi and I. Capuzzo Dolcetta, "Optimal Control and Viscosity Solutions of HamiltonJacobiBellman Equations," With appendices by Maurizio Falcone and Pierpaolo Soravia, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1997. 
[7] 
T. Başar and G. J. Olsder, "Dynamic Noncooperative Game Theory," Second edition, Academic Press, Ltd., London, 1995. 
[8] 
A. Bensoussan and J. Frehse, "Regularity Results for Nonlinear Elliptic Systems and Applications," Applied Mathematical Sciences, 151, SpringerVerlag, Berlin, 2002. 
[9] 
P. Cardaliaguet, "Notes on Mean Field Games," from P.L. Lions' lectures at Collège de France, 2010. 
[10] 
J. C. Engwerda, "Linear Quadratic Dynamic Optimization and Differential Games," Wiley, Chichester, 2005. 
[11] 
W. H. Fleming and H. M. Soner, "Controlled Markov Processes and Viscosity Solutions," 2^{nd} edition, Stochastic Modelling and Applied Probability, 25, Springer, New York, 2006. 
[12] 
D. A. Gomes, J. Mohr and R. R. Souza, Discrete time, finite state space mean field games, J. Math. Pures Appl. (9), 93 (2010), 308328. 
[13] 
O. Guéant, "Mean Field Games and Applications to Economics," Ph.D. Thesis, Université ParisDauphine, 2009. 
[14] 
O. Guéant, A reference case for mean field games models, J. Math. Pures Appl. (9), 92 (2009), 276294. 
[15] 
O. Guéant, J.M. Lasry and P.L. Lions, Mean field games and applications, in "ParisPrinceton Lectures on Mathematical Finance 2010" (eds. R. A. Carmona, et al.), Lecture Notes in Math., 2003, Springer, Berlin, (2011), 205266. 
[16] 
R. Z. Has'minskiĭ, "Stochastic Stability of Differential Equations," Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis, 7, Sijthoff & Noordhoff, Alphen aan den RijnGermantown, Md., 1980. 
[17] 
M. Huang, P. E. Caines and R. P. Malhamé, Individual and mass behaviour in large population stochastic wireless power control problems: Centralized and Nash equilibrium solutions, in "Proc. the 42^{nd} IEEE Conference on Decision and Control," Maui, Hawaii, December, (2003), 98103. 
[18] 
M. Huang, P. E. Caines and R. P. Malhamé, Large population stochastic dynamic games: Closedloop McKeanVlasov systems and the Nash certainty equivalence principle, Commun. Inf. Syst., 6 (2006), 221251. 
[19] 
M. Huang, P. E. Caines and R. P. Malhamé, Largepopulation costcoupled LQG problems with nonuniform agents: Individualmass behavior and decentralized $\epsilon$Nash equilibria, IEEE Trans. Automat. Control, 52 (2007), 15601571. doi: 10.1109/TAC.2007.904450. 
[20] 
M. Huang, P. E. Caines and R. P. Malhamé, An invariance principle in large population stochastic dynamic games, J. Syst. Sci. Complex., 20 (2007), 162172. doi: 10.1007/s1142400790154. 
[21] 
A. Lachapelle, J. Salomon and G. Turinici, Computation of mean field equilibria in economics, Math. Models Methods Appl. Sci., 20 (2010), 567588. doi: 10.1142/S0218202510004349. 
[22] 
J.M. Lasry and P.L. Lions, Jeux à champ moyen. I. Le cas stationnaire, C. R. Acad. Sci. Paris, 343 (2006), 619625. doi: 10.1016/j.crma.2006.09.019. 
[23] 
J.M. Lasry and P.L. Lions, Jeux à champ moyen. II. Horizon fini et contrôle optimal, C. R. Acad. Sci. Paris, 343 (2006), 679684. doi: 10.1016/j.crma.2006.09.018. 
[24] 
J.M. Lasry and P.L. Lions, Mean field games, Jpn. J. Math., 2 (2007), 229260. 
show all references
References:
[1] 
Y. Achdou, F. Camilli and I. CapuzzoDolcetta, Mean field games: Numerical methods for the planning problem, SIAM J. Control Opt., 50 (2012), 77109. doi: 10.1137/100790069. 
[2] 
Y. Achdou and I. CapuzzoDolcetta, Mean field games: Numerical methods, SIAM J. Numer. Anal., 48 (2010), 11361162. doi: 10.1137/090758477. 
[3] 
O. Alvarez and M. Bardi, Ergodic problems in differential games, in "Advances in Dynamic Game Theory," Ann. Internat. Soc. Dynam. Games, 9, Birkhäuser Boston, Boston, MA, (2007), 131152. 
[4] 
O. Alvarez and M. Bardi, Ergodicity, stabilization, and singular perturbations for BellmanIsaacs equations, Mem. Amer. Math. Soc., 204 (2010), vi+77 pp. 
[5] 
R. J. Aumann, Markets with a continuum of traders, Econometrica, 32 (1964), 3950. doi: 10.2307/1913732. 
[6] 
M. Bardi and I. Capuzzo Dolcetta, "Optimal Control and Viscosity Solutions of HamiltonJacobiBellman Equations," With appendices by Maurizio Falcone and Pierpaolo Soravia, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1997. 
[7] 
T. Başar and G. J. Olsder, "Dynamic Noncooperative Game Theory," Second edition, Academic Press, Ltd., London, 1995. 
[8] 
A. Bensoussan and J. Frehse, "Regularity Results for Nonlinear Elliptic Systems and Applications," Applied Mathematical Sciences, 151, SpringerVerlag, Berlin, 2002. 
[9] 
P. Cardaliaguet, "Notes on Mean Field Games," from P.L. Lions' lectures at Collège de France, 2010. 
[10] 
J. C. Engwerda, "Linear Quadratic Dynamic Optimization and Differential Games," Wiley, Chichester, 2005. 
[11] 
W. H. Fleming and H. M. Soner, "Controlled Markov Processes and Viscosity Solutions," 2^{nd} edition, Stochastic Modelling and Applied Probability, 25, Springer, New York, 2006. 
[12] 
D. A. Gomes, J. Mohr and R. R. Souza, Discrete time, finite state space mean field games, J. Math. Pures Appl. (9), 93 (2010), 308328. 
[13] 
O. Guéant, "Mean Field Games and Applications to Economics," Ph.D. Thesis, Université ParisDauphine, 2009. 
[14] 
O. Guéant, A reference case for mean field games models, J. Math. Pures Appl. (9), 92 (2009), 276294. 
[15] 
O. Guéant, J.M. Lasry and P.L. Lions, Mean field games and applications, in "ParisPrinceton Lectures on Mathematical Finance 2010" (eds. R. A. Carmona, et al.), Lecture Notes in Math., 2003, Springer, Berlin, (2011), 205266. 
[16] 
R. Z. Has'minskiĭ, "Stochastic Stability of Differential Equations," Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis, 7, Sijthoff & Noordhoff, Alphen aan den RijnGermantown, Md., 1980. 
[17] 
M. Huang, P. E. Caines and R. P. Malhamé, Individual and mass behaviour in large population stochastic wireless power control problems: Centralized and Nash equilibrium solutions, in "Proc. the 42^{nd} IEEE Conference on Decision and Control," Maui, Hawaii, December, (2003), 98103. 
[18] 
M. Huang, P. E. Caines and R. P. Malhamé, Large population stochastic dynamic games: Closedloop McKeanVlasov systems and the Nash certainty equivalence principle, Commun. Inf. Syst., 6 (2006), 221251. 
[19] 
M. Huang, P. E. Caines and R. P. Malhamé, Largepopulation costcoupled LQG problems with nonuniform agents: Individualmass behavior and decentralized $\epsilon$Nash equilibria, IEEE Trans. Automat. Control, 52 (2007), 15601571. doi: 10.1109/TAC.2007.904450. 
[20] 
M. Huang, P. E. Caines and R. P. Malhamé, An invariance principle in large population stochastic dynamic games, J. Syst. Sci. Complex., 20 (2007), 162172. doi: 10.1007/s1142400790154. 
[21] 
A. Lachapelle, J. Salomon and G. Turinici, Computation of mean field equilibria in economics, Math. Models Methods Appl. Sci., 20 (2010), 567588. doi: 10.1142/S0218202510004349. 
[22] 
J.M. Lasry and P.L. Lions, Jeux à champ moyen. I. Le cas stationnaire, C. R. Acad. Sci. Paris, 343 (2006), 619625. doi: 10.1016/j.crma.2006.09.019. 
[23] 
J.M. Lasry and P.L. Lions, Jeux à champ moyen. II. Horizon fini et contrôle optimal, C. R. Acad. Sci. Paris, 343 (2006), 679684. doi: 10.1016/j.crma.2006.09.018. 
[24] 
J.M. Lasry and P.L. Lions, Mean field games, Jpn. J. Math., 2 (2007), 229260. 
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