基于Dynkin对策和Dirichlet形式的多维随机奇异控制
Multi-dimensional Stochastic Singular Control Via Dynkin Game and Dirichlet Form
摘要：
随机奇异控制的传统难点在于刻画值函数的奇异性和最优控制策略。本文考虑了一个多维奇异控制问题。通过Dynkin对策得到了该问题的最优值函数和最优控制策略，其解由费用函数的鞍点给出。通过一个Dirichlet形式的变分不等式问题证明了该Dynkin对策解的存在唯一性。因此，Dynkin对策的值函数的性质意味着随机奇异控制问题的值函数的光滑性。通过这种方式，我们能够证明这个多维奇异控制问题的经典解的存在性，这是传统上解决的意义上的黏性解，这使得应用验证定理证明最优性。
英文摘要：
The traditional difficulty about stochastic singular control is to characterize the regularities of the value function and the optimal control policy. In this paper, a multi-dimensional singular control problem is considered. We found the optimal value function and the optimal control policy of this problem via Dynkin game, whose solution is given by the saddle point of the cost function. The existence and uniqueness of the solution to this Dynkin game are proved through an associated variational inequality problem involving Dirichlet form. As a consequence, the properties of the value function of this Dynkin game implies the smoothness of the value function of the stochastic singular control problem. In this way, we are able to show the existence of a classical solution to this multi-dimensional singular control problem, which was traditionally solved in the sense of viscosity solutions, and this enables the application of the verification theorem to prove optimality.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
