Annals of Probability

Mean field games with common noise

René Carmona, François Delarue, and Daniel Lacker

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Abstract

A theory of existence and uniqueness is developed for general stochastic differential mean field games with common noise. The concepts of strong and weak solutions are introduced in analogy with the theory of stochastic differential equations, and existence of weak solutions for mean field games is shown to hold under very general assumptions. Examples and counter-examples are provided to enlighten the underpinnings of the existence theory. Finally, an analog of the famous result of Yamada and Watanabe is derived, and it is used to prove existence and uniqueness of a strong solution under additional assumptions.

Article information

Source
Ann. Probab., Volume 44, Number 6 (2016), 3740-3803.

Dates
Received: July 2014
Revised: July 2015
First available in Project Euclid: 14 November 2016

Permanent link to this document
https://projecteuclid.org/euclid.aop/1479114262

Digital Object Identifier
doi:10.1214/15-AOP1060

Mathematical Reviews number (MathSciNet)
MR3572323

Zentralblatt MATH identifier
06674837

Subjects
Primary: 93E20: Optimal stochastic control
Secondary: 60H10: Stochastic ordinary differential equations [See also 34F05] 91A13: Games with infinitely many players

Keywords
Mean field games stochastic optimal control McKean–Vlasov equations weak solutions relaxed controls

Citation

Carmona, René; Delarue, François; Lacker, Daniel. Mean field games with common noise. Ann. Probab. 44 (2016), no. 6, 3740--3803. doi:10.1214/15-AOP1060. https://projecteuclid.org/euclid.aop/1479114262


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