标题： 控制的位移单调平均场博弈的定量收敛 显示英文标题

标题： Quantitative convergence for displacement monotone Mean Field Games of control

摘要： 在本文中，我们建立了在均值场控制博弈（MFGC）框架下，N人随机微分博弈的开环和闭环纳什均衡的定量收敛结果，这是一类代理之间的相互作用通过状态和控制同时发生的模型。 我们的分析涵盖了满足位移单调性条件的一般非可分离哈密顿量，以及在无穷远处的适度正则性和增长条件。 我们工作的重大创新之处在于对测度空间上的非平凡不动点问题进行了严格的处理，这在MFGC公式中自然出现。 与之前的工作不同，这些工作要么限制于可分离的哈密顿量——使不动点映射变得简单——要么假设不动点映射的收敛性或正则性，我们开发了对该方程及其N人对应物的详细结构分析。 这导致了不动点映射的新正则性结果，并进而导致了开环均衡的定量收敛。 我们进一步推导了N人纳什系统的精确先验估计，使我们能够控制开环和闭环策略之间的差异，从而得出闭环均衡的收敛性。 我们的框架也以自然的方式容纳了常见的噪声。 显示英文摘要

摘要： In this paper we establish quantitative convergence results for both open and closed-loop Nash equilibria of N-player stochastic differential games in the setting of Mean Field Games of Controls (MFGC), a class of models where interactions among agents occur through both states and controls. Our analysis covers a general class of non-separable Hamiltonians satisfying a displacement monotonicity condition, along with mild regularity and growth conditions at infinity. A major novelty of our work is the rigorous treatment of a nontrivial fixed-point problem on a space of measures, which arises naturally in the MFGC formulation. Unlike prior works that either restrict to separable Hamiltonians - rendering the fixed-point map trivial - or assume convergence or regularity properties of the fixed point map, we develop a detailed structural analysis of this equation and its N-player analogue. This leads to new regularity results for the fixed-point maps and, in turn, to quantitative convergence of open-loop equilibria. We further derive sharp a priori estimates for the N-player Nash system, enabling us to control the discrepancy between open and closed-loop strategies, and thus to conclude the convergence of closed-loop equilibria. Our framework also accommodates common noise in a natural way.