New stochastic verification theorems

This paper studies controlled systems governed by Ito´s stochastic differential equations in which control variables are allowed to enter both drift and diffusion terms. It turns out that verification theorems still hold if the derivatives of the value functions are replaced by any point in the super-/sub-differentials. These new verification theorems are shown to have wider applicability than the restrictive classical verification theorems which require the associated dynamic programming equations to have smooth solutions. Based on the new verification result, optimal stochastic feedback controls are obtained by maximizing the generalized Hamiltonians over both the control regions and the super-differentials of the value functions