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 stable model (SM) semantics lacks the properties of existence, relevance
and cumulativity. If we prospectively consider the class of conservative
extensions of SM semantics (i.e., semantics that for each normal logic program
P retrieve a superset of the set of stable models of P), one may wander how do
the semantics of this class behave in what concerns the aforementioned
properties. That is the type of issue dealt with in this paper. We define a
large class of conservative extensions of the SM semantics, dubbed affix stable
model semantics, ASM, and study the above referred properties into two
non-disjoint subfamilies of the class ASM, here dubbed ASMh and ASMm. From this
study a number of results stem which facilitate the assessment of semantics in
the class ASMh U ASMm with respect to the properties of existence, relevance
and cumulativity, whilst unveiling relations among these properties. As a
result of the approach taken in our work, light is shed on the characterization
of the SM semantics, as we show that the properties of (lack of) existence and
(lack of) cautious monotony are equivalent, which opposes statements on this
issue that may be found in the literature; we also characterize the relevance
failure of SM semantics in a more clear way than usually stated in the
literature.

本研究旨在探讨一类保守扩展下的稳定模型语义在存在性、相关性和累加性等属性方面的行为，并定义了这一类扩展中的一种大类，即转缀稳定模型语义，探究了其子类 ASMh 和 ASMm。通过研究，揭示了这些属性之间的关系。此外，我们还得出了一些结论，这些结论有助于评估 ASMh U ASMm 中的语义，同时揭示了 SM 语义的特征。机遇巧妙地证明了 SM 语义的存在性和谨慎单调性的特性是等价的，并对 SM 语义中的相关性失败进行了更清晰的描述。