Before revision: 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. A verification theorem concludes the resolutions of this problem .
After revision: By combining viscosity solutions with a change of measure technique introduced by Kuroda and Nagai we show that the problem reduces to solving a certain stochastic control problem in the factor process, which has no jumps. The main result of the paper is to show that the risk-sensitive jump diffusion problem can be fully characterized in terms of a parabolic Hamilton-Jacobi-Bellman PDE rather than a PIDE, and that this PDE admits a classical C^{1,2 .