Second Order Hamilton-Jacobi Equations in Hilbert Spaces and Stochastic Optimal Control

This paper is devoted to a brief survey of known results on second order Hamilton-Jacobi (HJ from now on) equations in Hilbert spaces related to stochastic optimal control problems. More precisely the paper focus on:

stating results of existence and uniqueness of regular solutions (i.e. at least Fréchet differentiable in the space variable) for semilinear second order HJ equation;

when the HJ equation is associated to a stochastic optimal control problem, applying these results to the associated stochastic optimal control problems to obtain verification theorems and feedback formulae for the optimal control strategies.