This paper investigates sufficient conditions for a Feynman-Kac functional up to an exit time to be the generalized viscosity solution of a Dirichlet problem. The key ingredient is to find out the continuity of an exit operator under the Skorokhod topology, which reveals the intrinsic connection between the overfitting Dirichlet boundary and fine topology. As an application, we establish the sub- and supersolutions for a class of nonstationary Hamilton-Jacobi-Bellman (HJB) equations with fractional Laplacian operator via Feynman-Kac functionals associated to alpha-stable processes, which lead to the existence of a strong solution to the original HJB equation.