Fokker-Planck equations of jumping particles and mean field games of impulse control
Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 5, pp. 1211-1244.
Le texte intégral des articles récents est réservé aux abonnés de la revue. Consultez l'article sur le site de la revue.

This paper is interested in the description of the density of particles evolving according to some optimal policy of an impulse control problem. We first fix the sets from which the particles jump and explain how we can characterize such a density. We then investigate the coupled case in which the underlying impulse control problem depends on the density we are looking for: the mean field game of impulse control. In both cases, we give a variational characterization of the densities of jumping particles.

DOI : 10.1016/j.anihpc.2020.04.006
Mots-clés : Partial differential equations, Fokker-Planck equations, Impulse control, Mean field games 
@article{AIHPC_2020__37_5_1211_0,
     author = {Bertucci, Charles},
     title = {Fokker-Planck equations of jumping particles and mean field games of impulse control},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1211--1244},
     publisher = {Elsevier},
     volume = {37},
     number = {5},
     year = {2020},
     doi = {10.1016/j.anihpc.2020.04.006},
     mrnumber = {4138232},
     zbl = {1456.49030},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2020.04.006/}
}
TY  - JOUR
AU  - Bertucci, Charles
TI  - Fokker-Planck equations of jumping particles and mean field games of impulse control
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2020
SP  - 1211
EP  - 1244
VL  - 37
IS  - 5
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.anihpc.2020.04.006/
DO  - 10.1016/j.anihpc.2020.04.006
LA  - en
ID  - AIHPC_2020__37_5_1211_0
ER  - 
%0 Journal Article
%A Bertucci, Charles
%T Fokker-Planck equations of jumping particles and mean field games of impulse control
%J Annales de l'I.H.P. Analyse non linéaire
%D 2020
%P 1211-1244
%V 37
%N 5
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.anihpc.2020.04.006/
%R 10.1016/j.anihpc.2020.04.006
%G en
%F AIHPC_2020__37_5_1211_0
Bertucci, Charles. Fokker-Planck equations of jumping particles and mean field games of impulse control. Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 5, pp. 1211-1244. doi : 10.1016/j.anihpc.2020.04.006. http://archive.numdam.org/articles/10.1016/j.anihpc.2020.04.006/

[1] Achdou, Yves; Capuzzo-Dolcetta, Italo Mean field games: numerical methods, SIAM J. Numer. Anal., Volume 48 (2010) no. 3, pp. 1136–1162 | MR | Zbl

[2] Achdou, Yves; Giraud, Pierre-Noel; Lasry, Jean-Michel; Lions, Pierre-Louis A long-term mathematical model for mining industries, Appl. Math. Optim., Volume 74 (2016) no. 3, pp. 579–618 | MR | Zbl

[3] Basei, Matteo; Cao, Haoyang; Guo, Xin Nonzero-sum stochastic games with impulse controls, 2019 (arXiv preprint) | arXiv

[4] Bensoussan, Alain; Lions, Jacques Louis Impulse Control and Quasi-Variational Inequalities, Gaunthier-Villars, 1984 | MR

[5] Bertucci, Charles Optimal stopping in mean field games, an obstacle problem approach, J. Math. Pures Appl. (2017) | MR | Zbl

[6] Briceno-Arias, L.M.; Kalise, D.; Silva, F.J. Proximal methods for stationary mean field games with local couplings, SIAM J. Control Optim., Volume 56 (2018) no. 2, pp. 801–836 | MR | Zbl

[7] Calvo, Juan; Novaga, Matteo; Orlandi, Giandomenico Parabolic equations in time-dependent domains, J. Evol. Equ., Volume 17 (2017) no. 2, pp. 781–804 | MR | Zbl

[8] Cardaliaguet, Pierre Notes on mean field games, 2010 (Technical report)

[9] Cardaliaguet, Pierre; Hadikhanloo, Saeed Learning in mean field games: the fictitious play, ESAIM Control Optim. Calc. Var., Volume 23 (2017) no. 2, pp. 569–591 | Numdam | MR | Zbl

[10] Cardaliaguet, Pierre; Porretta, Alessio Long time behavior of the master equation in mean-field game theory, 2017 (arXiv preprint) | arXiv

[11] Cardaliaguet, Pierre; Delarue, François; Lasry, Jean-Michel; Lions, Pierre-Louis The master equation and the convergence problem in mean field games, 2015 (arXiv preprint) | arXiv

[12] Cardaliaguet, Pierre; Jameson Graber, P.; Porretta, Alessio; Tonon, Daniela Second order mean field games with degenerate diffusion and local coupling, Nonlinear Differ. Equ. Appl., Volume 22 (2015) no. 5, pp. 1287–1317 | MR | Zbl

[13] Carmona, René; Delarue, François Probabilistic analysis of mean-field games, SIAM J. Control Optim., Volume 51 (2013) no. 4, pp. 2705–2734 | MR | Zbl

[14] Carmona, Rene; Delarue, François Probabilistic Theory of Mean Field Games with Applications I-II, Springer, 2017 | MR

[15] Carmona, Rene; Delarue, François; Lacker, Daniel Mean field games of timing and models for bank runs, Appl. Math. Optim., Volume 76 (2017) no. 1, pp. 217–260 | MR | Zbl

[16] Fu, Guanxing; Horst, Ulrich Mean field games with singular controls, SIAM J. Control Optim., Volume 55 (2017) no. 6, pp. 3833–3868 | MR | Zbl

[17] Gianazza, Ugo; Savaré, Giuseppe Abstract evolution equations on variable domains: an approach by minimizing movements, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 23 (1996) no. 1, pp. 149–178 | Numdam | MR | Zbl

[18] Gomes, Diogo; Patrizi, Stefania Obstacle mean-field game problem, Interfaces Free Bound., Volume 17 (2015) no. 1, pp. 55–68 | MR | Zbl

[19] Gomes, Diogo A.; Patrizi, Stefania Weakly coupled mean-field game systems, Nonlinear Anal., Volume 144 (2016), pp. 110–138 | MR | Zbl

[20] Guéant, Olivier A reference case for mean field games models, J. Math. Pures Appl., Volume 92 (2009) no. 3, pp. 276–294 | MR | Zbl

[21] Guéant, Olivier; Lasry, Jean-Michel; Lions, Pierre-Louis Mean field games and applications, Paris-Princeton Lectures on Mathematical Finance 2010, Springer, 2011, pp. 205–266 | MR | Zbl

[22] Guo, Xin; Lee, Joon Seok Mean field games with singular controls of bounded velocity, 2017 (arXiv preprint) | arXiv

[23] Huang, Minyi; Malhamé, Roland P.; Caines, Peter E. Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle, Commun. Inf. Syst., Volume 6 (2006) no. 3, pp. 221–252 | MR | Zbl

[24] Kakutani, Shizuo A generalization of Brouwer's fixed point theorem, Duke Math. J., Volume 8 (1941) no. 3, pp. 457–459 | MR | Zbl

[25] Krein, Mark Grigor'evich; Rutman, Moisei Aronovich Linear operators leaving invariant a cone in a Banach space, Usp. Mat. Nauk, Volume 3 (1948) no. 1, pp. 3–95 | MR | Zbl

[26] Lacker, Daniel Mean field games via controlled martingale problems: existence of Markovian equilibria, Stoch. Process. Appl., Volume 125 (2015) no. 7, pp. 2856–2894 | MR | Zbl

[27] Laetsch, Theodore A uniqueness theorem for elliptic quasi-variational inequalities, J. Funct. Anal., Volume 18 (1975) no. 3, pp. 286–287 | MR | Zbl

[28] Lasry, Jean-Michel; Lions, Pierre-Louis Jeux à champ moyen. I–le cas stationnaire, C. R. Math., Volume 343 (2006) no. 9, pp. 619–625 | MR | Zbl

[29] Lasry, Jean-Michel; Lions, Pierre-Louis Jeux à champ moyen. II–horizon fini et contrôle optimal, C. R. Math., Volume 343 (2006) no. 10, pp. 679–684 | MR | Zbl

[30] Lasry, Jean-Michel; Lions, Pierre-Louis Mean field games, Jpn. J. Math., Volume 2 (2007) no. 1, pp. 229–260 | MR | Zbl

[31] Lions, Pierre-Louis Cours au college de france, 2011 www.college-de-france.fr (2007)

[32] Nutz, Marcel A mean field game of optimal stopping, 2016 (arXiv preprint) | arXiv | MR

[33] Yaegashi, Yuta; Yoshioka, Hidekazu; Tsugihashi, Kentaro; Fujihara, Masayuki Analysis and computation of probability density functions for a 1-d impulsively controlled diffusion process, C. R. Math., Volume 357 (2019) no. 3, pp. 306–315 | MR | Zbl

Cité par Sources :