扩散过程的双边遍历奇异控制与平均场博弈
Two sided ergodic singular control and mean field game for diffusions
AI总结：
本文研究线性扩散驱动的平均场博弈，先求解无相互作用的双边遍历奇异控制问题，证明双阈值反射策略最优，并据此建立反射型总体控制下均衡存在判据、非线性方程及有限Nash均衡的逼近结果。
摘要：
在一个由线性扩散驱动的概率平均场博弈中，单个参与者试图通过一对递增和递减的càdlàg过程控制该扩散，以最小化遍历长期成本；同时，他通过另一个由一对càdlàg过程控制的类似扩散的期望，与参与者总体发生相互作用。为了寻找该博弈的均衡点，我们首先考虑单个参与者与参与者总体没有相互作用的控制问题。在这种情形下，我们证明最优策略是将扩散过程反射在两个阈值之内。基于这些结果，在参与者总体的控制属于反射类型的情况下，我们得到了平均场博弈中均衡点存在性的判据，并给出一对用于寻找这些均衡点的非线性方程。此外，当参与者数量趋于无穷大时，我们给出了有限参与者遍历博弈的Nash均衡向上述平均场博弈均衡逼近的结果。这些结果通过若干例子加以说明，在这些例子中，均衡点的存在性和唯一性取决于基础扩散的系数。
英文摘要：
In a probabilistic mean-field game driven by a linear diffusion an individual player aims to minimize an ergodic long-run cost by controlling the diffusion through a pair of -- increasing and decreasing -- càdlàg processes, while he is interacting with an aggregate of players through the expectation of a similar diffusion controlled by another pair of càdlàg processes. In order to find equilibrium points in this game, we first consider the control problem, in which the individual player has no interaction with the aggregate of players. In this case, we prove that the best policy is to reflect the diffusion process within two thresholds. Based on these results, we obtain criteria for the existence of equilibrium points in the mean-field game in the case when the controls of the aggregate of players are of reflection type, and give a pair of nonlinear equations to find these equilibrium points. In addition, we present an approximation result for Nash equilibria of erdogic games with finitely many players to the mean-field game equilibria considered above when the number of players tends to infinity. These results are illustrated by several examples where the existence and uniqueness of the equilibrium points depend on the coefficients of the underlying diffusion.
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