AI 中文总结
本文针对通过松弛受控鞅问题构建的随机平均场博弈引入可容许颤抖手精炼，证明其均衡存在性，并通过一维模型验证该精炼可选择性识别颤抖手完美均衡。
AI 中文摘要
我们为通过松弛受控鞅问题构建的随机平均场博弈（MFG）引入了一种可容许的颤抖手精炼。若松弛MFG均衡是两组联合控制-状态律序列的共同Wasserstein极限，则称其为完美均衡：第一组由针对其生成的状态律具有全支撑的总体扰动构成；第二组是对这些扰动状态律的精确最优响应。该可容许性条件防止扰动的控制与状态坐标独立变化。在标准连续性、增长性与强制性假设下，我们无需紧致控制或有界系数即可证明存在性。证明结合了紧致有界情形下的扰动不动点论证，以及截断、一致矩估计、紧致性与受控鞅问题的稳定性。一个具有恰好两个松弛平均场博弈均衡的一维模型表明，该精炼具有真正的选择性：仅一个均衡是颤抖手完美的。
英文摘要
We introduce an admissible trembling-hand refinement for stochastic mean field games formulated through relaxed controlled martingale problems. A relaxed MFG equilibrium is perfect if it is the common Wasserstein limit of two sequences of joint control-state laws. The first consists of full-support population perturbations admissible for the state laws they generate; the second consists of exact optimal responses to those perturbed state laws. This admissibility condition prevents the control and state coordinates of a perturbation from being varied independently. Under the standing continuity, growth, and coercivity assumptions, we prove existence without requiring compact controls or bounded coefficients. The proof combines a perturbed fixed-point argument in the compact-bounded case with truncation, uniform moment estimates, compactness, and stability of controlled martingale problems. A one-dimensional model with exactly two relaxed mean field game equilibria shows that the refinement is genuinely selective: only one equilibrium is trembling-hand perfect.
