论文标题
平均现场游戏主方程：从离散到连续状态空间
Mean field games master equations: from discrete to continuous state space
论文作者
论文摘要
本文研究了具有有限状态空间的平均现场游戏的融合，即表示具有连续状态空间的现场游戏。我们检查了扩散动力学的空间离散化，该动力学让人联想到马尔可夫链近似方法中的固有控制方法，但也想起有限的差异数值方案。时间在离散化中保持连续，并且时间范围任意长。我们主要对关联的主方程的解决方案的融合感兴趣，因为状态数量倾向于无穷大。我们提出了两种方法，用于在单调性假设下以无噪声或共同噪声处理案例。第一个使用主方程的特性系统（即MFG系统）来建立无常见噪声和相关最佳轨迹的主方程的收敛速率，以防万一对极限主方程有一个平稳的解决方案，并且没有。第二种方法取决于Bertucci引入的单调溶液的概念。在存在共同噪声的情况下，我们显示主方程的收敛性，如果限制主方程平滑，则具有收敛速率，否则通过紧凑型参数。
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; time remains continuous in the discretization, and the time horizon is arbitrarily long. 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.
