含粗糙公共噪声的均场博弈：紧致化方法
Mean-field games with rough common noise: the compactification approach
AI总结：
本文研究含粗糙公共噪声的均场博弈问题，通过引入松弛控制和粗糙鞅问题的弱形式，证明了路径均场均衡的存在性，并讨论了路径问题与经典随机布朗运动公共噪声的均场博弈之间的关系。
摘要：
我们研究了含粗糙公共噪声的均场博弈（MFG）问题，其中代表性状态动态由受异质布朗运动和确定性粗糙路径信号驱动的受控粗糙随机微分方程所支配。在此新框架中，我们引入了基于松弛控制和粗糙鞅问题的规范弱形式。我们通过开发新的紧致化工具来证明路径均场均衡的存在性，这些工具能够处理粗糙积分，并与文献中的经典紧致化论证有显著不同。最后，我们讨论了路径问题与经典带有随机布朗运动公共噪声的均场博弈问题之间的关系。利用路径可接受集的概念，我们将带有公共噪声的均场博弈问题转化为扩展概率测度空间上的优化问题。我们建立了Carmona-Delarue-Lacker弱均衡的等价刻画，并作为应用，给出了强均衡的另一种证明方式，而无需首先证明路径唯一性。
英文摘要：
We study mean-field game (MFG) problems with rough common noise, in which the representative state dynamics are governed by a controlled rough stochastic differential equation driven by an idiosyncratic Brownian motion and a deterministic rough-path signal that affects the whole population. Within this new framework, we introduce a canonical weak formulation based on relaxed controls and rough martingale problems. We prove the existence of a pathwise mean-field equilibrium by developing new compactification tools that accommodate rough integration and differ substantially from classical compactification arguments in the literature. Finally, we discuss the relationship between the pathwise problem and the classical MFG problem with randomized Brownian common noise. Using the notion of a pathwise admissible set, we recast mean-field game problems with common noise as optimization problems over an extended space of probability measures. We establish an equivalent characterization of Carmona-Delarue-Lacker's weak equilibrium and, as an application, give an alternative proof of strong equilibrium without first establishing pathwise uniqueness.
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