Monotone solutions for mean field games master equations: finite state space and optimal stopping
[Solutions monotones des équations maîtresses des jeux à champ moyen]
Journal de l’École polytechnique — Mathématiques, Tome 8 (2021) , pp. 1099-1132.

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.

Reçu le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/jep.167
Classification : 35Q89,  35F50,  91A16
Mots clés : Équations aux dérivées partielles, jeux à champ moyen
@article{JEP_2021__8__1099_0,
     author = {Charles Bertucci},
     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 = {https://jep.centre-mersenne.org/articles/10.5802/jep.167/}
}
Charles Bertucci. 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. https://jep.centre-mersenne.org/articles/10.5802/jep.167/

[1] Y. Achdou & I. Capuzzo-Dolcetta - “Mean field games: numerical methods”, SIAM J. Numer. Anal. 48 (2010) no. 3, p. 1136-1162 | Article | MR 2679575 | Zbl 1217.91019

[2] Y. Achdou & M. Laurière - “Mean field games and applications: numerical aspects”, 2020 | arXiv:2003.04444

[3] L. Briceño-Arias, D. Kalise, Z. Kobeissi, M. Laurière, Á. Mateos González & F. J. Silva - “On the implementation of a primal-dual algorithm for second order time-dependent mean field games with local couplings”, in CEMRACS 2017—Numerical methods for stochastic models: control, uncertainty quantification, mean-field, ESAIM Proc. Surveys, vol. 65, EDP Sciences, Les Ulis, 2019, p. 330-348 | Article | MR 3968547 | Zbl 1418.49032

[4] E. Bayraktar, A. Cecchin, A. Cohen & F. Delarue - “Finite state mean field games with Wright-Fisher common noise”, 2019 | arXiv:1912.06701

[5] E. Bayraktar & A. Cohen - “Analysis of a finite state many player game using its master equation”, SIAM J. Control Optimization 56 (2018) no. 5, p. 3538-3568 | Article | MR 3860894 | Zbl 1416.91013

[6] A. Bensoussan & J. L. Lions - Impulse control and quasi-variational inequalities, Gauthier-Villars, Paris, 1984

[7] C. Bertucci - “Optimal stopping in mean field games, an obstacle problem approach”, J. Math. Pures Appl. (9) 120 (2018), p. 165-194 | Article | MR 3906158 | Zbl 1406.35448

[8] C. Bertucci - “Fokker-Planck equations of jumping particles and mean field games of impulse control”, Ann. Inst. H. Poincaré Anal. Non Linéaire 37 (2020) no. 5, p. 1211-1244 | Article | MR 4138232 | Zbl 1456.49030

[9] C. Bertucci - “A remark on Uzawa’s algorithm and an application to mean field games systems”, ESAIM Math. Model. Numer. Anal. 54 (2020) no. 3, p. 1053-1071 | Article | MR 4094733 | Zbl 1437.91051

[10] C. Bertucci - “Work in progress”, 2021

[11] C. Bertucci, L. Bertucci, J.-M. Lasry & P.-L. Lions - “Mean field game approach to bitcoin mining”, 2020 | arXiv:2004.08167

[12] C. Bertucci, J.-M. Lasry & P.-L. Lions - “Some remarks on mean field games”, Comm. Partial Differential Equations 44 (2019) no. 3, p. 205-227 | Article | MR 3941633 | Zbl 1411.91100

[13] J. M. Borwein & D. Noll - “Second order differentiability of convex functions in Banach spaces”, Trans. Amer. Math. Soc. 342 (1994) no. 1, p. 43-81 | Article | MR 1145959 | Zbl 0802.46027

[14] P. Cardaliaguet, F. Delarue, J.-M. Lasry & P.-L. Lions - The master equation and the convergence problem in mean field games, Annals of Math. Studies, vol. 201, Princeton University Press, Princeton, NJ, 2019 | Article | MR 3967062 | Zbl 1430.91002

[15] P. Cardaliaguet, P. J. Graber, A. Porretta & D. Tonon - “Second order mean field games with degenerate diffusion and local coupling”, NoDEA Nonlinear Differential Equations Appl. 22 (2015) no. 5, p. 1287-1317 | Article | MR 3399179 | Zbl 1344.49061

[16] P. Cardaliaguet, J.-M. Lasry, P.-L. Lions & A. Porretta - “Long time average of mean field games”, Netw. Heterog. Media 7 (2012) no. 2, p. 279-301 | Article | MR 2928380 | Zbl 1270.35098

[17] P. Cardaliaguet & A. Porretta - “Long time behavior of the master equation in mean field game theory”, Anal. PDE 12 (2019) no. 6, p. 1397-1453 | Article | MR 3921309 | Zbl 1428.35607

[18] R. Carmona, F. Delarue & et al. - Probabilistic theory of mean field games with applications. I-II, Probability Theory and Stochastic Modelling, vol. 83 & 84, Springer, Cham, 2018

[19] R. Carmona, F. Delarue & D. Lacker - “Mean field games of timing and models for bank runs”, Appl. Math. Optim. 76 (2017) no. 1, p. 217-260 | Article | MR 3679343 | Zbl 1411.91102

[20] A. Cecchin & G. Pelino - “Convergence, fluctuations and large deviations for finite state mean field games via the master equation”, Stochastic Processes Appl. 129 (2019) no. 11, p. 4510-4555 | Article | MR 4013871 | Zbl 1450.60015

[21] A. Cecchin, P. D. Pra, M. Fischer & G. Pelino - “On the convergence problem in mean field games: a two state model without uniqueness”, SIAM J. Control Optimization 57 (2019) no. 4, p. 2443-2466 | Article | MR 3981375 | Zbl 1426.91025

[22] J. Claisse, Z. Ren & X. Tan - “Mean field games with branching”, 2019 | arXiv:1912.11893

[23] M. G. Crandall & P.-L. Lions - “Viscosity solutions of Hamilton-Jacobi equations”, Trans. Amer. Math. Soc. 277 (1983) no. 1, p. 1-42 | Article | MR 690039 | Zbl 0599.35024

[24] M. Fabian & C. Finet - “On Stegall’s smooth variational principle”, Nonlinear Anal. 66 (2007) no. 3, p. 565-570 | Article | MR 2274868 | Zbl 1117.46014

[25] W. Gangbo & A. R. Mészáros - “Global well-posedness of master equations for deterministic displacement convex potential mean field games”, 2020 | arXiv:2004.01660

[26] W. Gangbo, A. R. Mészáros, C. Mou & J. Zhang - “Mean field games master equations with non-separable Hamiltonians and displacement monotonicity”, 2021 | arXiv:2101.12362

[27] D. A. Gomes & S. Patrizi - “Weakly coupled mean-field game systems”, Nonlinear Anal. 144 (2016), p. 110-138 | Article | MR 3534097 | Zbl 1358.35200

[28] D. Lacker - “A general characterization of the mean field limit for stochastic differential games”, Probab. Theory Related Fields 165 (2016) no. 3-4, p. 581-648 | Article | MR 3520014 | Zbl 1344.60065

[29] D. Lacker - “On the convergence of closed-loop Nash equilibria to the mean field game limit”, 2018 | arXiv:1808.02745

[30] J.-M. Lasry & P.-L. Lions - “Une classe nouvelle de problèmes singuliers de contrôle stochastique”, C. R. Acad. Sci. Paris Sér. I Math. 331 (2000) no. 11, p. 879-885 | Article | Zbl 0971.49015

[31] J.-M. Lasry & P.-L. Lions - “Mean field games”, Japan. J. Math. (N.S.) 2 (2007) no. 1, p. 229-260 | Article | MR 2295621 | Zbl 1156.91321

[32] P.-L. Lions - “Cours au Collège de France”, 2011, http://www.college-de-france.fr

[33] P.-L. Lions - “Cours au Collège de France”, 2018, http://www.college-de-france.fr

[34] C. Mou & J. Zhang - “Wellposedness of second order master equations for mean field games with nonsmooth data”, 2019 | arXiv:1903.09907

[35] M. Nutz - “A mean field game of optimal stopping”, SIAM J. Control Optimization 56 (2018) no. 2, p. 1206-1221 | Article | MR 3780736 | Zbl 1407.91040

[36] M. Nutz, J. San Martin & X. Tan - “Convergence to the mean field game limit: a case study”, Ann. Appl. Probab. 30 (2020) no. 1, p. 259-286 | Article | MR 4068311 | Zbl 1437.91058

[37] O. A. Oleĭnik & E. V. Radkevič - Second order equations with nonnegative characteristic form, American Mathematical Society, Providence, RI, and Plenum Press, New York-London, 1973 | Article

[38] A. Porretta - “Weak solutions to Fokker–Planck equations and mean field games”, Arch. Rational Mech. Anal. 216 (2015) no. 1, p. 1-62 | Article | MR 3305653 | Zbl 1312.35168

[39] C. Stegall - “Optimization of functions on certain subsets of Banach spaces”, Math. Ann. 236 (1978) no. 2, p. 171-176 | Article | MR 503448 | Zbl 0365.49006

[40] C. Stegall - “Optimization and differentiation in Banach spaces”, Linear Algebra and Appl. 84 (1986), p. 191-211 | Article | MR 872283 | Zbl 0633.46042