Mean field games.

*(English)*Zbl 1156.91321Summary: We survey here some recent studies concerning what we call mean-field models by analogy with statistical mechanics and physics. More precisely, we present three examples of our mean-field approach to modelling in economics and finance (or other related subjects …). Roughly speaking, we are concerned with situations that involve a very large number of “rational players” with a limited information (or visibility) on the “game”. Each player chooses his optimal strategy in view of the global (or macroscopic) informations that are available to him and that result from the actions of all players. In the three examples we mention here, we derive a mean-field problem which consists in nonlinear differential equations. These equations are of a new type and our main goal here is to study them and establish their links with various fields of analysis. We show in particular that these nonlinear problems are essentially well-posed problems i.e., have unique solutions. In addition, we give various limiting cases, examples and possible extensions. And we mention many open problems.

##### MSC:

91A16 | Mean field games (aspects of game theory) |

91A23 | Differential games (aspects of game theory) |

82B05 | Classical equilibrium statistical mechanics (general) |

91A15 | Stochastic games, stochastic differential games |

91G80 | Financial applications of other theories |

35J99 | Elliptic equations and elliptic systems |

35K99 | Parabolic equations and parabolic systems |

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\textit{J.-M. Lasry} and \textit{P.-L. Lions}, Jpn. J. Math. (3) 2, No. 1, 229--260 (2007; Zbl 1156.91321)

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