Before revision:  Building upon results in the first stage, we construct a sequence of approximating risk sensitive control problems with finite state space and prove that the resulting smooth value functions will converge to the classical solution of the original system of DPEs.
After revision:  Building upon results in the first stage, we construct a sequence of approximating risk sensitive control problems with finite states and prove that the resulting smooth value functions will converge to the classical solution of the original system of DPEs.