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December 2018 Verification theorems for stochastic optimal control problems in Hilbert spaces by means of a generalized Dynkin formula
Salvatore Federico, Fausto Gozzi
Ann. Appl. Probab. 28(6): 3558-3599 (December 2018). DOI: 10.1214/18-AAP1397

Abstract

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.

Citation

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Salvatore Federico. Fausto Gozzi. "Verification theorems for stochastic optimal control problems in Hilbert spaces by means of a generalized Dynkin formula." Ann. Appl. Probab. 28 (6) 3558 - 3599, December 2018. https://doi.org/10.1214/18-AAP1397

Information

Received: 1 February 2017; Revised: 1 December 2017; Published: December 2018
First available in Project Euclid: 8 October 2018

zbMATH: 06994400
MathSciNet: MR3861820
Digital Object Identifier: 10.1214/18-AAP1397

Subjects:
Primary: 49L20, 49N35, 65H15, 70H20, 93E20

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.28 • No. 6 • December 2018
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