Optimal Control of Stochastic Integrals and Hamilton–Jacobi–Bellman Equations. II

We consider the solution of a stochastic integral control problem and we study its regularity. In particular, we characterize the optimal cost as the maximum solution of \[{gathered} ∀ v ∈ V, A(v)u f(v) {in }D'(O), \\ u = 0 {on }∂ O, u ∈ W1,∞ (O), \\ {gathered} \] where $A(v)$ is a uniformly elliptic second order operator and V is the set of the values of the control.