Stochastic Verification Theorems within the Framework of Viscosity Solutions

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. A new verification theorem is derived within the framework of viscosity solutions without involving any derivatives of the value functions. This theorem is 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 superdifferentials of the value functions.