Verification by Stochastic Perronʹs Method in stochastic exit time control problems

We apply the Stochastic Perron Method, created by Bayraktar and Sîrbu, to a stochastic exit time control problem. Our main assumption is the validity of the Strong Comparison Result for the related Hamilton–Jacobi–Bellman (HJB) equation. Without relying on Bellmanʹs optimality principle we prove that inside the domain the value function is continuous and coincides with a viscosity solution of the Dirichlet boundary value problem for the HJB equation.