We formulate and analyze a mathematical framework for continuous-time mean field games with finitely many states and common noise, including a rigorous probabilistic construction of the state process and existence and uniqueness results for the resulting equilibrium system. The key insight is that we can circumvent the master equation and reduce the mean field equilibrium to a system of forward-backward systems of (random) ordinary differential equations by conditioning on common noise events. In the absence of common noise, our setup reduces to that of Gomes, Mohr and Souza (Appl Math Optim 68(1): 99–143, 2013) and Cecchin and Fischer (Appl Math Optim 81(2):253–300, 2020).

我们为具有有限状态和共同噪声的连续时间均值场博弈制定并分析了数学框架，包括状态过程的严格概率构造以及所得均衡系统的存在和唯一性结果。关键的见解是，我们可以通过限制常见的噪声事件，来规避主方程，并将平均场平衡降低为（随机）常微分方程的前-后系统。在没有常见噪声的情况下，我们的设置将简化为Gomes，Mohr和Souza（Appl Math Optim 68（1）：99-143，2013年）和Cecchin和Fischer（Appl Math Optim 81（2）：253-300， 2020）。