[Solutions monotones des équations maîtresses des jeux à champ moyen]
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
Charles Bertucci 1
CC-BY 4.0
@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/}
}
TY - JOUR AU - Charles Bertucci 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 - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.167/ DO - 10.5802/jep.167 LA - en ID - JEP_2021__8__1099_0 ER -
%0 Journal Article %A Charles Bertucci %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 École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.167/ %R 10.5802/jep.167 %G en %F JEP_2021__8__1099_0
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/
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