## Annals of Probability

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

### Mean field games with common noise

René Carmona, François Delarue, and Daniel Lacker

#### 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