This paper studies the convergence of mean field games with finite state space to mean field games with a continuous state space. We examine a space discretization of a diffusive dynamics, which is reminiscent of the Markov chain approximation method in stochasctic control, but also of finite difference numerical schemes. We are mainly interested in the convergence of the solution of the associated master equations as the number of states tends to infinity. We present two approaches, to treat the case without or with common noise, both under monotonicity assumptions. The first one uses the system of characteristics of the master equation, which is the MFG system, to establish a convergence rate for the master equations without common noise and the associated optimal trajectories, both in case there is a smooth solution to the limit master equation and in case there is not. The second approach relies on the notion of monotone solutions introduced by Bertucci. In the presence of common noise, we show convergence of the master equations, with a convergence rate if the limit master equation is smooth, otherwise by compactness arguments.
翻译:本文研究与有限状态空间相近的平均野外游戏的趋同性, 是指与连续状态空间相交的野外游戏。 我们研究的是, diffusive 动态的空间分解, 它在随机查控中与Markov链近似法相近, 但也与有限差异数值方案相类似。 我们主要关心的是, 相关主方程的解决方案的趋同性, 因为国家数量往往不尽相同。 我们提出了两种方法, 即根据单一度假设, 不使用或以共同噪音处理案件。 首先, 我们使用主方程的特性系统, 即MFG 系统, 来为主方程设定一种趋同率, 而没有常见噪音, 以及相关的最佳轨迹。 第二种方法依赖于Bertcucci提出的单项解决方案的概念。 在存在常见的噪音的情况下, 我们展示了母方程的趋同性, 如果总方程的限度是平坦的, 则以压缩度参数为趋同率 。