Mean field games with common noise. (English) Zbl 1422.91083

Carmona, René; Delarue, François; Lacker, Daniel

Ann. Probab. 44, No. 6, 3740-3803 (2016); erratum ibid. 48, No. 5, 2644-2646 (2020).

Summary: 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 T. Yamada and S. Watanabe [J. Math. Kyoto Univ. 11, 155–167 (1971; Zbl 0236.60037)] is derived, and it is used to prove existence and uniqueness of a strong solution under additional assumptions.

MSC:

91A15	Stochastic games, stochastic differential games
91A23	Differential games (aspects of game theory)
93E20	Optimal stochastic control
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)

Keywords:

mean field games; stochastic optimal control; McKean-Vlasov equations; weak solutions; relaxed controls

Citations:

Zbl 0236.60037