Annals of Probability

Mean field games with common noise

René Carmona, François Delarue, and Daniel Lacker

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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