Optimal control of nonlinear stochastic differential equations on Hilbert spaces

We here consider optimal control problems governed by nonlinear stochastic equations on a Hilbert space H with nonconvex payoff, which is rewritten as a deterministic optimal control problem governed by a Kolmogorov equation in H. We prove the existence and first-order necessary condition of closed loop optimal controls for the above control problem. The strategy is based on solving a deterministic bilinear optimal control problem for the corresponding Kolmogorov equation on the space $L^2(H,\nu)$, where $\nu$ is the related infinitesimally invariant measure for the Kolmogorov operator.