标题： 带有共同噪声的线性二次均场博弈：一种直接方法 显示英文标题

标题： Linear-Quadratic Mean Field Games with Common Noise: A Direct Approach

摘要： 本文研究了一个具有共同噪声的线性二次均值场博弈问题，其中个体状态方程的漂移项和扩散项与状态、控制和均值场项耦合，我们采用直接方法来解决这个问题。 与处理相应的均值场团队问题相比，具有状态耦合的均值场博弈问题更具挑战性。 这不仅体现在应用变分分析时伴随方程数量的爆炸性增长，还体现在解耦高维前向-后向随机微分方程系统时需要更多的Riccati方程。 我们采取了一套不同的步骤，并巧妙地利用了方程的固有性质来解决这一挑战。 首先，我们在一个大规模且有限的群体设定中求解一个$N$-玩家博弈问题，并通过变分分析得到一组前向-后向随机微分方程。 然后，我们通过取$N$趋于无穷大并应用大数定律来推导极限前向-后向随机微分方程。 基于后向随机微分方程解的存在性和唯一性，方程中的一些变量恒等于零，这显著降低了分析的复杂性。 这使我们能够引入两个Riccati方程来显式构造所有参与者的去中心化策略。 此外，我们证明了所构建的去中心化策略构成原始问题的$\epsilon$-纳什均衡策略。 我们还将结果扩展到无限时间范围的情况，并分析代数Riccati方程的可解性。 最后，提供了数值模拟以说明上述结论。 显示英文摘要

摘要： This paper investigates a linear-quadratic mean field games problem with common noise, where the drift term and diffusion term of individual state equations are coupled with both the state, control, and mean field terms of the state, and we adopt the direct approach to tackle this problem. Compared with addressing the corresponding mean field teams problem, the mean field games problem with state coupling presents greater challenges. This is not only reflected in the explosive increase in the number of adjoint equations when applying variational analysis but also in the need for more Riccati equations during decoupling the high-dimensional forward-backward stochastic differential equations system. We take a different set of steps and ingeniously utilize the inherent properties of the equations to address this challenge. First, we solve an $N$-player games problem within a vast and finite population setting, and obtain a set of forward-backward stochastic differential equations by variational analysis. Then, we derive the limiting forward-backward stochastic differential equations by taking the limit as $N$ approaches infinity and applying the law of large numbers. Based on the existence and uniqueness of solutions to backward stochastic differential equations, some variables in the equations are identically zero, which significantly reduces the complexity of the analysis. This allows us to introduce just two Riccati equations to explicitly construct decentralized strategies for all participants. Moreover, we demonstrate that the constructed decentralized strategies constitute an $\epsilon$-Nash equilibrium strategy for the original problem. We also extend the results to the infinite-horizon case and analyze the solvability of algebraic Riccati equations. Finally, numerical simulations are provided to illustrate the preceding conclusions.