VERIFICATION THEOREMS FOR STOCHASTIC OPTIMAL CONTROL PROBLEMS IN HILBERT SPACES BY MEANS OF A GENERALIZED DYNKIN FORMULA

• 成果类型: Article
• 署名作者: Federico, Salvatore; Gozzi, Fausto
• 署名单位: University of Siena; Luiss Guido Carli University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/18-AAP1397
• 发表日期: 2018
• 页码: 3558-3599
• 关键词: hamilton-jacobi equations; differential-equations; continuous semigroups; elliptic-equations; invariant-measures; delay; DIFFUSIONS; horizon

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 (in the sense of [J. Differential Equations 262 (2017) 3343-3389]). The main methodological novelty of our result relies on the fact that it is not needed to prove, as in previous literature (see, e.g., [Comm. Partial Differential Equations 20 (1995) 775-826]), that the mild solution is a strong solution, that is, a suitable limit of classical solutions of approximating HJB equations. To achieve the goal, we prove a new type of Dynkin formula, which is the key tool for the proof of our main result.