希尔伯特空间中随机最优控制问题的推广Dynkin公式验证定理
Verification theorems for stochastic optimal control problems in Hilbert spaces by means of a generalized Dynkin formula
摘要：
验证定理是动态规划方法成功应用于最优控制问题的关键结果。本文介绍一种新的方法来证明无穷维随机最优控制问题的验证定理。该方法适用于附加控制的Ornstein-Uhlenbeck过程的情况下，当相关的Hamilton-Jacobi-Bellman（HJB）方程承认一个温和的解决方案。我们的结果的主要方法的新颖性依赖于这样一个事实，它是不需要证明，如在以前的文献中，温和的解决方案是一个强解，即一个适当的限制经典解的HJB方程。为了实现我们的目标，我们证明了一种新的Dynkin公式，这是证明我们的主要结果的关键工具。
英文摘要：
Verification theorems are key results to successfully employ the dynamic programming approach to optimal control problems. In this paper we introduce a new method to prove verification theorems for infinite dimensional stochastic optimal control problems. The method applies in the case of additively controlled Ornstein-Uhlenbeck processes, when the associated Hamilton-Jacobi-Bellman (HJB) equation admits a mild solution. The main methodological novelty of our result relies on the fact that it is not needed to prove, as in previous literature, that the mild solution is a strong solution, i.e. a suitable limit of classical solutions of the HJB equation. To achieve our goal we prove a new type of Dynkin formula, which is the key tool for the proof of our main result.
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