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A semi-discrete approximation for a first order mean field game problem

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  • In this article we consider a model first order mean field game problem, introduced by J.M. Lasry and P.L. Lions in [18]. Its solution $(v,m)$ can be obtained as the limit of the solutions of the second order mean field game problems, when the noise parameter tends to zero (see [18]). We propose a semi-discrete in time approximation of the system and, under natural assumptions, we prove that it is well posed and that it converges to $(v,m)$ when the discretization parameter tends to zero.
    Mathematics Subject Classification: Primary: 91A13; Secondary: 65M25, 49L25.

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  • [1]

    Y. Achdou, F. Camilli and I. Capuzzo Dolcetta, Mean field games: Numerical methods for the planning problem, SIAM J. of Control & Optimization, 50 (2012), 77-109.doi: 10.1137/100790069.

    [2]

    Y. Achdou and I. Capuzzo-Dolcetta, Mean field games: Numerical methods, SIAM J. Numer. Anal., 48 (2010), 1136-1162.doi: 10.1137/090758477.

    [3]

    J.-P. Aubin and H. Frankowska, "Set-Valued Analysis," Systems & Control: Foundations & Applications, 2, Birkhäuser Boston, Inc., Boston, MA, 1990.

    [4]

    M. Bardi and I. Capuzzo-Dolcetta, "Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations," With appendices by Maurizio Falcone and Pierpaolo Soravia, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1997.

    [5]

    J. Bonnans and A. Shapiro, "Perturbation Analysis of Optimization Problems," Springer Series in Operations Research, Springer-Verlag, New York, 2000.

    [6]

    P. Cannarsa and C. Sinestrari, "Semiconcave functions, Hamilton-Jacobi equations, and Optimal Control," Progress in Nonlinear Differential Equations and their Applications, 58, Birkhäuser Boston, Inc., Boston, MA, 2004.

    [7]

    Pierre Cardaliaguet, "Notes on Mean Field Games: From P.-L. Lions' Lectures at Collège de France," Lecture Notes given at Tor Vergata, 2010.

    [8]

    I. Capuzzo-Dolcetta, On a discrete approximation of the Hamilton-Jacobi equation of dynamic programming, Appl. Math. Optim., 10 (1983), 367-377.doi: 10.1007/BF01448394.

    [9]

    I. Capuzzo-Dolcetta and M. Falcone, Discrete dynamic programming and viscosity solutions of the Bellman equation, Analyse Non Linéaire (Perpignan, 1987), Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 161-183.

    [10]

    I. Capuzzo-Dolcetta and H. Ishii, Approximate solutions of the Bellman equation of deterministic control theory, Appl. Math. Optim., 11 (1984), 161-181.doi: 10.1007/BF01442176.

    [11]

    M. Falcone and R. Ferretti, "Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations," MOS-SIAM Series on Optimization, to appear.

    [12]

    D. A. Gomes, Viscosity solution methods and the discrete Aubry-Mather problem, Discrete Contin. Dyn. Syst., 13 (2005), 103-116.doi: 10.3934/dcds.2005.13.103.

    [13]

    D. A. Gomes, J. Mohr and R. R. Souza, Discrete time, finite state space mean field games, J. Math. Pures Appl. (9), 93 (2010), 308-328.

    [14]

    O. Guéant, Mean field games equations with quadratic hamiltonian: a specific approach, arXiv:1106.3269v1, 2011.

    [15]

    A. Lachapelle, J. Salomon and G. Turinici, Computation of mean field equilibria in economics, Math. Models Methods Appl. Sci., 20 (2010), 567-588.doi: 10.1142/S0218202510004349.

    [16]

    J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. I. Le cas stationnaire, C. R. Math. Acad. Sci. Paris, 343 (2006), 619-625.doi: 10.1016/j.crma.2006.09.019.

    [17]

    J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. II. Horizon fini et contrôle optimal, C. R. Math. Acad. Sci. Paris, 343 (2006), 679-684.doi: 10.1016/j.crma.2006.09.019.

    [18]

    J.-M. Lasry and P.-L. Lions, Mean field games, Jpn. J. Math., 2 (2007), 229-260.

    [19]

    P.-L. Lions, Cours du Collège de France. Available from: http://www.college-de-france.fr.

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