Classical solutions of the Backward PIDE for a Marked Point Processes with characteristics modulated by a jump diffusion

The objective of this paper is to give conditions ensuring that the backward partial integro differential equation (PIDE) arising from a multidimensional jump-diffusion with a pure jump component has a classical solution, that is the solution is continuous, $\mathcal C^2$ in the diffusion component and $\mathcal{C}^1$ in time. Our proof uses a probabilistic arguments and extends the results of Pham (1998) to the case where the diffusion operator is not elliptic in all components and where the jump intensity is modulated by a diffusion process.