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We study optimal control problems governed by abstract infinite dimensionalstochastic differential equations using the dynamic programming approach. Inthe first part, we prove Lipschitz continuity, semiconcavity and semiconvexityof 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, weconstruct optimal feedback controls using the notion of the $B$-continuousviscosity solutions for the associated Hamilton--Jacobi--Bellman equation. Thisis done in the case when the noise coefficient is independent of the controlvariable. We also discuss applications of our results to optimal controlproblems for controlled stochastic reaction-diffusion equations and, undereconomic motivations, to stochastic delay differential equations.

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