AI 中文总结
本文研究有限时间平均场奇异随机控制问题，通过势平均场博弈方法求解经典单调追随者问题，首次完整刻画该类问题的最优策略。
AI 中文摘要
我们研究一类有限时间范围内带奇异控制的平均场控制（MFC）问题，允许成本泛函对测度自变量具有一般依赖关系。我们推导得到一个带奇异控制的辅助平均场博弈（MFG），将其称为势MFG，并证明在适当凸性假设下，该势MFG的任意解都可得到原MFC问题的解。我们将这一一般结果应用于I. Karatzas和S. E. Shreve（《SIAM控制与优化期刊》22卷6期，第856-877页，1984年）提出的经典单调追随者问题的一个带标量平均场交互的版本。通过利用其与优化步骤最优停止的关联，并恰当应用Kakutani-Fan-Glicksberg不动点定理，求解了对应的带奇异控制的势MFG。在策略互补情形下，平均场均衡（即原MFC问题的最优策略）由一个连续非增自由边界刻画，该自由边界唯一求解一个非线性积分方程。据我们所知，本文是首个完整刻画有限时间平均场奇异随机控制问题最优策略的研究。
英文摘要
We study a class of mean-field control (MFC) problems with singular controls over a finite horizon, allowing for general dependence of the cost functional on the measure argument. We derive an auxiliary mean-field game (MFG) with singular controls, which we refer to as a potential MFG, and show that, under suitable convexity assumptions, any solution to this potential MFG yields a solution to the original MFC problem. We apply this general result to a version of the classical Monotone Follower Problem by I. Karatzas and S. E. Shreve (SIAM Journal on Control and Optimization 22(6), pp. 856-877, 1984) with scalar mean-field interaction. The associated potential MFG with singular controls is solved by exploiting its connection with optimal stopping for the optimization step and by a suitable application of the Kakutani-Fan-Glicksberg fixed-point theorem. In the case of strategic complementarities, the mean-field equilibrium (and hence the optimal policy of the original MFC problem) is characterized by a continuous nonincreasing free boundary that uniquely solves a nonlinear integral equation. To the best of our knowledge, this is the first paper to provide a complete characterization of the optimal policy in a finite-horizon mean-field singular stochastic control problem.
