In this paper, we consider Mean Field Games in the presence of common noise
relaxing the usual independence assumption of individual random noise. We
assume a simple linear model with terminal cost satisfying a convexity and a
weak monotonicity property. Our main result is showing existence and uniqueness
of a Mean Field Game solution using the Stochastic Maximum Principle. The
uniqueness is a result of a monotonicity property similar to that of Lasry and
Lions. We use the Banach Fixed Point Theorem to establish an existence over
small time duration and show that it can be extended over an arbitrary finite
time duration.

本文探讨了在具有共同噪声的情况下的平均场博弈，使用随机最大化原理证明了存在唯一的平均场博弈解，并证明了其具有类似于 Lasry 和 Lions 的单调性质，使用 Banach 不动点定理证明了其在有限时间内的存在性，并展示其可以延伸至任意有限时间内。

具有共同噪声的均值场博弈的良好性满足弱单调性条件

Wellposedness of Mean Field Games with Common Noise Under a Weak  Monotonicity Condition

The purpose of this paper is to provide a detailed probabilistic analysis of
the optimal control of nonlinear stochastic dynamical systems of the McKean
Vlasov type. Motivated by the recent interest in mean field games, we highlight
the connection and the differences between the two sets of problems. We prove a
new version of the stochastic maximum principle and give sufficient conditions
for existence of an optimal control. We also provide examples for which our
sufficient conditions for existence of an optimal solution are satisfied.
Finally we show that our solution to the control problem provides approximate
equilibria for large stochastic games with mean field interactions.

通过概率分析，研究了 McKeanVlasov 型非线性随机动力系统的最优控制问题，给出了最优解的充分条件，并将其应用于含有均场交互的大规模随机博弈中。