Stochastic optimal control in Hilbert spaces: $C^{1,1}$ regularity of the value function and optimal synthesis via viscosity solutions

TL;DR

This work addresses stochastic optimal control problems in infinite-dimensional Hilbert spaces driven by unbounded operators. It develops $C^{1,1}$ regularity of the value function in the state variable by proving semiconcavity and semiconvexity under several scenarios, enabling robust regularity results even when diffusion is degenerate or the state operator is unbounded. For the semilinear case, the authors construct optimal feedback controls via $B$-continuous viscosity solutions of the Hamilton–Jacobi–Bellman equation, bypassing the need for classical Itô verification by leveraging a linear Kolmogorov equation and a Feynman–Kac representation. The paper also extends these results to the weak $B$-condition and demonstrates applications to controlled stochastic reaction–diffusion and delay equations, including new SPDE comparison results that may be of independent interest. Overall, it provides a rigorous pathway from regularity of the value function to explicit optimal synthesis in infinite dimensions, with broad applicability to SPDEs and SDDEs.

Abstract

We study optimal control problems governed by abstract infinite dimensional stochastic differential equations using the dynamic programming approach. In the first part, we prove Lipschitz continuity, semiconcavity and semiconvexity of the value function under several sets of assumptions, and thus derive its $C^{1,1}$ regularity in the space variable. Based on this regularity result, we construct optimal feedback controls using the notion of the $B$-continuous viscosity solutions for the associated Hamilton--Jacobi--Bellman equation. This is done in the case when the noise coefficient is independent of the control variable. We also discuss applications of our results to optimal control problems governed by stochastic reaction-diffusion equations and, under economic motivations, stochastic delay differential equations.

Theorems & Definitions (56)

• Remark 2.1
• Definition 2.4
• Remark 2.5
• Definition 2.6
• Definition 2.7
• Lemma 3.3
• proof
• Remark 3.5
• Theorem 3.6
• proof