多维奇异控制及相关 Skorokhod 问题：刻画最优控制的充分条件
Multidimensional singular control and related Skorokhod problem: sufficient conditions for the characterization of optimal controls
AI总结：
本文将多维奇异随机控制问题的最优控制刻画为相关 Skorokhod 反射问题的唯一解，通过研究价值函数导数的单调性构造近最优策略并取极限，为退化和非退化设定及线性二次模型提供了最优控制构造的充分条件。
摘要：
我们将一类奇异随机控制问题的最优控制刻画为相关 Skorokhod 反射问题的唯一解。所考虑的优化问题涉及在无限时间跨度上，通过影响 Itô 扩散的有界变差过程来最小化折扣成本泛函。该设定是多维的，状态方程的漂移项和成本是凸的，波动率矩阵可以是常数或状态的线性函数。我们证明，最优控制仅在底层扩散试图离开所谓的等待区域时才起作用，且该作用的方向由价值函数的导数规定。我们的方法基于通过将价值函数导数解释为最优停止博弈的值，来研究其适当的单调性。这种单调性允许构造近最优策略，该策略在近似等待区域的边界上反射底层扩散。然后，该近似方案的极限提供了所需的刻画。除其他外，我们的结果适用于一类相关的线性二次模型。此外，它允许在文献中考虑的退化和非退化设定中构造最优控制，而这一重要方面此前仅得到部分解决。
英文摘要：
We characterize the optimal control for a class of singular stochastic control problems as the unique solution to a related Skorokhod reflection problem. The considered optimization problems concern the minimization of a discounted cost functional over an infinite time-horizon through a process of bounded variation affecting an Itô-diffusion. The setting is multidimensional, the drift of the state equation and the costs are convex, the volatility matrix can be constant or linear in the state. We prove that the optimal control acts only when the underlying diffusion attempts to exit the so-called waiting region, and that the direction of this action is prescribed by the derivative of the value function. Our approach is based on the study of a suitable monotonicity property of the derivative of the value function through its interpretation as the value of an optimal stopping game. Such a monotonicity allows to construct nearly optimal policies which reflect the underlying diffusion at the boundary of approximating waiting regions. The limit of this approximation scheme then provides the desired characterization. Our result applies to a relevant class of linear-quadratic models, among others. Furthermore, it allows to construct the optimal control in degenerate and non degenerate settings considered in the literature, where this important aspect was only partially addressed.
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