Verification by stochastic Perron's method in stochastic exit time\n control problems

We apply the Stochastic Perron method, created by Bayraktar and S\\^irbu, to a\nstochastic exit time control problem. Our main assumption is the validity of\nthe Strong Comparison Result for the related Hamilton-Jacobi-Bellman (HJB)\nequation. Without relying on Bellman's optimality principle we prove that\ninside the domain the value function is continuous and coincides with a\nviscosity solution of the Dirichlet boundary value problem for the HJB\nequation.\n