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

Filippo de Feo, Andrzej Święch, Lukas Wessels

Published 2023-10-04Version 1

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 solution of the associated Hamilton--Jacobi--Bellman equation. This is done in the case when the diffusion coefficient is independent of the control variable.