# American Institute of Mathematical Sciences

June  2012, 7(2): 315-336. doi: 10.3934/nhm.2012.7.315

## New numerical methods for mean field games with quadratic costs

 1 UFR de Math, Universit, 175, rue du Chevaleret, 75013 Paris, France

Received  November 2011 Revised  March 2012 Published  June 2012

Mean field games have been introduced by J.-M. Lasry and P.-L. Lions in [13, 14, 15] as the limit case of stochastic differential games when the number of players goes to $+\infty$. In the case of quadratic costs, we present two changes of variables that allow to transform the mean field games (MFG) equations into two simpler systems of equations. The first change of variables, introduced in [11], leads to two heat equations with nonlinear source terms. The second change of variables, which is introduced for the first time in this paper, leads to two Hamilton-Jacobi-Bellman equations. Numerical methods based on these equations are presented and numerical experiments are carried out.
Citation: Olivier Guéant. New numerical methods for mean field games with quadratic costs. Networks and Heterogeneous Media, 2012, 7 (2) : 315-336. doi: 10.3934/nhm.2012.7.315
##### References:
 [1] Y. Achdou, F. Camilli and I. Capuzzo-Dolcetta, Mean field games: Numerical methods for the planning problem, SIAM J. Control Opt., 50 (2012), 77-109. doi: 10.1137/100790069. [2] Y. Achdou and I. Capuzzo-Dolcetta, Mean field games: Numerical methods, SIAM Journal on Numerical Analysis, 48 (2010), 1136-1162. doi: 10.1137/090758477. [3] P. Cardaliaguet, Notes on mean field games, from P.-L. Lions' lectures at Collège de France, 2010. [4] M. G. Crandall and P.-L. Lions, Two approximations of solutions of Hamilton-Jacobi equations, Mathematics of Computation, 43 (1984), 1-19. doi: 10.1090/S0025-5718-1984-0744921-8. [5] L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, Vol. 19, American Mathematical Society, Providence, RI, 2010. [6] D. A. Gomes, J. Mohr and R. R. Souza, Discrete time, finite state space mean field games, Journal de Mathématiques Pures et Appliquées (9), 93 (2010), 308-328. [7] O. Guéant, Mean field games equations with quadratic hamiltonian: A specifc approach, to appear in Mathematical Models and Methods in Applied Sciences (M3AS). [8] O. Guéant, Mean field games with quadratic hamiltonian: A constructive scheme, to appear in the Annals of ISDG. [9] O. Guéant, "Mean Field Games and Applications to Economics," Ph.D thesis, Université Paris-Dauphine, 2009. [10] O. Guéant, A reference case for mean field games models, Journal de Mathématiques Pures et Appliquées (9), 92 (2009), 276-294. [11] O. Guéant, J.-M. Lasry and P.-L. Lions, Mean field games and applications, in "Paris Princeton Lectures on Mathematical Finance 2010," Lecture Notes in Math., 2003, Springer, Berlin, (2011), 205-266. [12] A. Lachapelle, J. Salomon and G. Turinici, Computation of mean field equilibria in economics, Mathematical Models and Methods in Applied Sciences, 20 (2010), 567-588. doi: 10.1142/S0218202510004349. [13] J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. I. Le cas stationnaire, C. R. Acad. Sci. Paris, 343 (2006), 619-625. doi: 10.1016/j.crma.2006.09.019. [14] J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. II. Horizon fini et contrôle optimal, C. R. Acad. Sci. Paris, 343 (2006), 679-684. doi: 10.1016/j.crma.2006.09.018. [15] J.-M. Lasry and P.-L. Lions, Mean field games, Japanese Journal of Mathematics, 2 (2007), 229-260. [16] P.-L. Lions, Théorie des jeux à champs moyens, Cours au Collège de France. Available from: http://www.college-de-france.fr/default/EN/all/equ_der/audio_video.jsp.

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##### References:
 [1] Y. Achdou, F. Camilli and I. Capuzzo-Dolcetta, Mean field games: Numerical methods for the planning problem, SIAM J. Control Opt., 50 (2012), 77-109. doi: 10.1137/100790069. [2] Y. Achdou and I. Capuzzo-Dolcetta, Mean field games: Numerical methods, SIAM Journal on Numerical Analysis, 48 (2010), 1136-1162. doi: 10.1137/090758477. [3] P. Cardaliaguet, Notes on mean field games, from P.-L. Lions' lectures at Collège de France, 2010. [4] M. G. Crandall and P.-L. Lions, Two approximations of solutions of Hamilton-Jacobi equations, Mathematics of Computation, 43 (1984), 1-19. doi: 10.1090/S0025-5718-1984-0744921-8. [5] L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, Vol. 19, American Mathematical Society, Providence, RI, 2010. [6] D. A. Gomes, J. Mohr and R. R. Souza, Discrete time, finite state space mean field games, Journal de Mathématiques Pures et Appliquées (9), 93 (2010), 308-328. [7] O. Guéant, Mean field games equations with quadratic hamiltonian: A specifc approach, to appear in Mathematical Models and Methods in Applied Sciences (M3AS). [8] O. Guéant, Mean field games with quadratic hamiltonian: A constructive scheme, to appear in the Annals of ISDG. [9] O. Guéant, "Mean Field Games and Applications to Economics," Ph.D thesis, Université Paris-Dauphine, 2009. [10] O. Guéant, A reference case for mean field games models, Journal de Mathématiques Pures et Appliquées (9), 92 (2009), 276-294. [11] O. Guéant, J.-M. Lasry and P.-L. Lions, Mean field games and applications, in "Paris Princeton Lectures on Mathematical Finance 2010," Lecture Notes in Math., 2003, Springer, Berlin, (2011), 205-266. [12] A. Lachapelle, J. Salomon and G. Turinici, Computation of mean field equilibria in economics, Mathematical Models and Methods in Applied Sciences, 20 (2010), 567-588. doi: 10.1142/S0218202510004349. [13] J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. I. Le cas stationnaire, C. R. Acad. Sci. Paris, 343 (2006), 619-625. doi: 10.1016/j.crma.2006.09.019. [14] J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. II. Horizon fini et contrôle optimal, C. R. Acad. Sci. Paris, 343 (2006), 679-684. doi: 10.1016/j.crma.2006.09.018. [15] J.-M. Lasry and P.-L. Lions, Mean field games, Japanese Journal of Mathematics, 2 (2007), 229-260. [16] P.-L. Lions, Théorie des jeux à champs moyens, Cours au Collège de France. Available from: http://www.college-de-france.fr/default/EN/all/equ_der/audio_video.jsp.
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