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.
René Carmona. François Delarue. Daniel Lacker. "Mean field games with common noise." Ann. Probab. 44 (6) 3740 - 3803, November 2016. https://doi.org/10.1214/15-AOP1060