  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.
