On présente une nouvelle notion de solution pour les équations maîtresses des jeux à champ moyen. Cette notion permet de travailler avec des fonctions qui sont simplement continues. On prouve en premier lieu des résultats d’unicité, d’existence et de stabilité pour de telles solutions. On montre alors dans la deuxième partie de cet article que cette notion permet de caractériser la fonction valeur de jeux à champ moyen d’arrêt optimal ou de contrôle impulsionnel. Cette notion a surtout un intérêt dans le cas monotone. On se restreint ici au cas d’un espace d’états fini.
We present a new notion of solution for mean field games master equations. This notion allows us to work with solutions which are merely continuous. We first prove results of uniqueness and stability for such solutions. It turns out that this notion is helpful to characterize the value function of mean field games of optimal stopping or impulse control and this is the topic of the second half of this paper. The notion of solution we introduce is only useful in the monotone case. In this article we focus on the finite state space case.
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Keywords: Partial differential equations, mean field games
Mot clés : Équations aux dérivées partielles, jeux à champ moyen
@article{JEP_2021__8__1099_0, author = {Bertucci, Charles}, title = {Monotone solutions for mean field games master equations: finite state space and optimal stopping}, journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques}, pages = {1099--1132}, publisher = {Ecole polytechnique}, volume = {8}, year = {2021}, doi = {10.5802/jep.167}, language = {en}, url = {http://www.numdam.org/articles/10.5802/jep.167/} }
TY - JOUR AU - Bertucci, Charles TI - Monotone solutions for mean field games master equations: finite state space and optimal stopping JO - Journal de l’École polytechnique — Mathématiques PY - 2021 SP - 1099 EP - 1132 VL - 8 PB - Ecole polytechnique UR - http://www.numdam.org/articles/10.5802/jep.167/ DO - 10.5802/jep.167 LA - en ID - JEP_2021__8__1099_0 ER -
%0 Journal Article %A Bertucci, Charles %T Monotone solutions for mean field games master equations: finite state space and optimal stopping %J Journal de l’École polytechnique — Mathématiques %D 2021 %P 1099-1132 %V 8 %I Ecole polytechnique %U http://www.numdam.org/articles/10.5802/jep.167/ %R 10.5802/jep.167 %G en %F JEP_2021__8__1099_0
Bertucci, Charles. Monotone solutions for mean field games master equations: finite state space and optimal stopping. Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 1099-1132. doi : 10.5802/jep.167. http://www.numdam.org/articles/10.5802/jep.167/
[1] Mean field games: numerical methods, SIAM J. Numer. Anal., Volume 48 (2010) no. 3, pp. 1136-1162 | DOI | MR | Zbl
[2] Mean field games and applications: numerical aspects, 2020 | arXiv
[3] On the implementation of a primal-dual algorithm for second order time-dependent mean field games with local couplings, CEMRACS 2017—Numerical methods for stochastic models: control, uncertainty quantification, mean-field (ESAIM Proc. Surveys), Volume 65, EDP Sciences, Les Ulis, 2019, pp. 330-348 | DOI | MR | Zbl
[4] Finite state mean field games with Wright-Fisher common noise, 2019 | arXiv
[5] Analysis of a finite state many player game using its master equation, SIAM J. Control Optimization, Volume 56 (2018) no. 5, pp. 3538-3568 | DOI | MR | Zbl
[6] Impulse control and quasi-variational inequalities, Gauthier-Villars, Paris, 1984
[7] Optimal stopping in mean field games, an obstacle problem approach, J. Math. Pures Appl. (9), Volume 120 (2018), pp. 165-194 | DOI | MR | Zbl
[8] Fokker-Planck equations of jumping particles and mean field games of impulse control, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 37 (2020) no. 5, pp. 1211-1244 | DOI | MR | Zbl
[9] A remark on Uzawa’s algorithm and an application to mean field games systems, ESAIM Math. Model. Numer. Anal., Volume 54 (2020) no. 3, pp. 1053-1071 | DOI | MR | Zbl
[10] Work in progress, 2021
[11] Mean field game approach to bitcoin mining, 2020 | arXiv
[12] Some remarks on mean field games, Comm. Partial Differential Equations, Volume 44 (2019) no. 3, pp. 205-227 | DOI | MR | Zbl
[13] Second order differentiability of convex functions in Banach spaces, Trans. Amer. Math. Soc., Volume 342 (1994) no. 1, pp. 43-81 | DOI | MR | Zbl
[14] The master equation and the convergence problem in mean field games, Annals of Math. Studies, 201, Princeton University Press, Princeton, NJ, 2019 | DOI | MR | Zbl
[15] Second order mean field games with degenerate diffusion and local coupling, NoDEA Nonlinear Differential Equations Appl., Volume 22 (2015) no. 5, pp. 1287-1317 | DOI | MR | Zbl
[16] Long time average of mean field games, Netw. Heterog. Media, Volume 7 (2012) no. 2, pp. 279-301 | DOI | MR | Zbl
[17] Long time behavior of the master equation in mean field game theory, Anal. PDE, Volume 12 (2019) no. 6, pp. 1397-1453 | DOI | MR | Zbl
[18] et al. Probabilistic theory of mean field games with applications. I-II, Probability Theory and Stochastic Modelling, 83 & 84, Springer, Cham, 2018
[19] Mean field games of timing and models for bank runs, Appl. Math. Optim., Volume 76 (2017) no. 1, pp. 217-260 | DOI | MR | Zbl
[20] Convergence, fluctuations and large deviations for finite state mean field games via the master equation, Stochastic Processes Appl., Volume 129 (2019) no. 11, pp. 4510-4555 | DOI | MR | Zbl
[21] On the convergence problem in mean field games: a two state model without uniqueness, SIAM J. Control Optimization, Volume 57 (2019) no. 4, pp. 2443-2466 | DOI | MR | Zbl
[22] Mean field games with branching, 2019 | arXiv
[23] Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., Volume 277 (1983) no. 1, pp. 1-42 | DOI | MR | Zbl
[24] On Stegall’s smooth variational principle, Nonlinear Anal., Volume 66 (2007) no. 3, pp. 565-570 | DOI | MR | Zbl
[25] Global well-posedness of master equations for deterministic displacement convex potential mean field games, 2020 | arXiv
[26] Mean field games master equations with non-separable Hamiltonians and displacement monotonicity, 2021 | arXiv
[27] Weakly coupled mean-field game systems, Nonlinear Anal., Volume 144 (2016), pp. 110-138 | DOI | MR | Zbl
[28] A general characterization of the mean field limit for stochastic differential games, Probab. Theory Related Fields, Volume 165 (2016) no. 3-4, pp. 581-648 | DOI | MR | Zbl
[29] On the convergence of closed-loop Nash equilibria to the mean field game limit, 2018 | arXiv
[30] Une classe nouvelle de problèmes singuliers de contrôle stochastique, C. R. Acad. Sci. Paris Sér. I Math., Volume 331 (2000) no. 11, pp. 879-885 | DOI | Zbl
[31] Mean field games, Japan. J. Math. (N.S.), Volume 2 (2007) no. 1, pp. 229-260 | DOI | MR | Zbl
[32] Cours au Collège de France, 2011 (http://www.college-de-france.fr)
[33] Cours au Collège de France, 2018 (http://www.college-de-france.fr)
[34] Wellposedness of second order master equations for mean field games with nonsmooth data, 2019 | arXiv
[35] A mean field game of optimal stopping, SIAM J. Control Optimization, Volume 56 (2018) no. 2, pp. 1206-1221 | DOI | MR | Zbl
[36] Convergence to the mean field game limit: a case study, Ann. Appl. Probab., Volume 30 (2020) no. 1, pp. 259-286 | DOI | MR | Zbl
[37] Second order equations with nonnegative characteristic form, American Mathematical Society, Providence, RI, and Plenum Press, New York-London, 1973 | DOI
[38] Weak solutions to Fokker–Planck equations and mean field games, Arch. Rational Mech. Anal., Volume 216 (2015) no. 1, pp. 1-62 | DOI | MR | Zbl
[39] Optimization of functions on certain subsets of Banach spaces, Math. Ann., Volume 236 (1978) no. 2, pp. 171-176 | DOI | MR | Zbl
[40] Optimization and differentiation in Banach spaces, Linear Algebra and Appl., Volume 84 (1986), pp. 191-211 | DOI | MR | Zbl
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