Before revision:  In this case, the HJB equation is a PIDE.By combining viscosity solutions with a change of notation, a policy improvement argument and classical results on parabolic PDEs we prove that the PIDE admits a unique smooth solution.
After revision:  In this case, the HJB equation is a change of notation, a policy improvement argument and classical results on parabolic PDEs we prove that the PIDE admits a unique smooth solution.