  We apply the Stochastic Perron method, created by Bayraktar and S\^irbu, 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 Bellman's 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.
