Before revision:  As an application, we establish the sub and supersolutions for a class of non-stationary HJB (Hamilton-Jacobi-Bellman) equations with fractional Laplacian operator via Feynman-Kac functionals associated to \alpha-stable processes, which enables us to verify its solvability together with comparison principle and Perron's method .
After revision:  As an application, we establish the sub and supersolutions for a class of non-stationary HJB (Hamilton-Jacobi-Bellman) equations with fractional Laplacian operator via Feynman-Kac functionals associated to \alpha-stable processes, which help verify the solvability of the original HJB equation .