# Verification theorems for stochastic optimal control problems in Hilbert spaces by means of a generalized Dynkin formula

## TL;DR

This paper introduces a new approach to verification theorems in infinite-dimensional stochastic control, utilizing a generalized Dynkin formula that relaxes the need for strong solution assumptions.

## Contribution

The paper presents a novel method for verification theorems in stochastic control that does not require the mild solution to be a strong solution, using a new Dynkin formula.

## Key findings

- Developed a new Dynkin formula for infinite-dimensional stochastic processes.
- Proved verification theorems without requiring strong solutions.
- Applicable to Ornstein-Uhlenbeck processes with mild solutions.

## 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. 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.