New numerical methods for mean field games with quadratic costs

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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

Abstract
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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.

Keywords: Hopf-Cole transformation., finite difference methods, numerical methods, constructive scheme, Mean field games.

Mathematics Subject Classification: Primary: 35K91, 65M12; Secondary: 91A1.

Citation: Olivier Guéant. New numerical methods for mean field games with quadratic costs. Networks & 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. doi: 10.1137/100790069. Google Scholar
[2]	Y. Achdou and I. Capuzzo-Dolcetta, Mean field games: Numerical methods,, SIAM Journal on Numerical Analysis, 48 (2010), 1136. doi: 10.1137/090758477. Google Scholar
[3]	P. Cardaliaguet, Notes on mean field games,, from P.-L. Lions' lectures at Collège de France, (2010). Google Scholar
[4]	M. G. Crandall and P.-L. Lions, Two approximations of solutions of Hamilton-Jacobi equations,, Mathematics of Computation, 43 (1984), 1. doi: 10.1090/S0025-5718-1984-0744921-8. Google Scholar
[5]	L. C. Evans, "Partial Differential Equations,", Graduate Studies in Mathematics, (2010). Google Scholar
[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. Google Scholar
[7]	O. Guéant, Mean field games equations with quadratic hamiltonian: A specifc approach,, to appear in Mathematical Models and Methods in Applied Sciences (M3AS)., (). Google Scholar
[8]	O. Guéant, Mean field games with quadratic hamiltonian: A constructive scheme,, to appear in the Annals of ISDG., (). Google Scholar
[9]	O. Guéant, "Mean Field Games and Applications to Economics,", Ph.D thesis, (2009). Google Scholar
[10]	O. Guéant, A reference case for mean field games models,, Journal de Mathématiques Pures et Appliquées (9), 92 (2009), 276. Google Scholar
[11]	O. Guéant, J.-M. Lasry and P.-L. Lions, Mean field games and applications,, in, 2003 (2011), 205. Google Scholar
[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. doi: 10.1142/S0218202510004349. Google Scholar
[13]	J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. I. Le cas stationnaire,, C. R. Acad. Sci. Paris, 343 (2006), 619. doi: 10.1016/j.crma.2006.09.019. Google Scholar
[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. doi: 10.1016/j.crma.2006.09.018. Google Scholar
[15]	J.-M. Lasry and P.-L. Lions, Mean field games,, Japanese Journal of Mathematics, 2 (2007), 229. Google Scholar
[16]	P.-L. Lions, Théorie des jeux à champs moyens, Cours au Collège de France., Available from: \url{http://www.college-de-france.fr/default/EN/all/equ_der/audio_video.jsp}., (). Google Scholar

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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. doi: 10.1137/100790069. Google Scholar
[2]	Y. Achdou and I. Capuzzo-Dolcetta, Mean field games: Numerical methods,, SIAM Journal on Numerical Analysis, 48 (2010), 1136. doi: 10.1137/090758477. Google Scholar
[3]	P. Cardaliaguet, Notes on mean field games,, from P.-L. Lions' lectures at Collège de France, (2010). Google Scholar
[4]	M. G. Crandall and P.-L. Lions, Two approximations of solutions of Hamilton-Jacobi equations,, Mathematics of Computation, 43 (1984), 1. doi: 10.1090/S0025-5718-1984-0744921-8. Google Scholar
[5]	L. C. Evans, "Partial Differential Equations,", Graduate Studies in Mathematics, (2010). Google Scholar
[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. Google Scholar
[7]	O. Guéant, Mean field games equations with quadratic hamiltonian: A specifc approach,, to appear in Mathematical Models and Methods in Applied Sciences (M3AS)., (). Google Scholar
[8]	O. Guéant, Mean field games with quadratic hamiltonian: A constructive scheme,, to appear in the Annals of ISDG., (). Google Scholar
[9]	O. Guéant, "Mean Field Games and Applications to Economics,", Ph.D thesis, (2009). Google Scholar
[10]	O. Guéant, A reference case for mean field games models,, Journal de Mathématiques Pures et Appliquées (9), 92 (2009), 276. Google Scholar
[11]	O. Guéant, J.-M. Lasry and P.-L. Lions, Mean field games and applications,, in, 2003 (2011), 205. Google Scholar
[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. doi: 10.1142/S0218202510004349. Google Scholar
[13]	J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. I. Le cas stationnaire,, C. R. Acad. Sci. Paris, 343 (2006), 619. doi: 10.1016/j.crma.2006.09.019. Google Scholar
[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. doi: 10.1016/j.crma.2006.09.018. Google Scholar
[15]	J.-M. Lasry and P.-L. Lions, Mean field games,, Japanese Journal of Mathematics, 2 (2007), 229. Google Scholar
[16]	P.-L. Lions, Théorie des jeux à champs moyens, Cours au Collège de France., Available from: \url{http://www.college-de-france.fr/default/EN/all/equ_der/audio_video.jsp}., (). Google Scholar

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