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The objective of this paper is to give conditions ensuring that the backwardpartial integro differential equation associated with a multidimensionaljump-diffusion with a pure jump component has a unique classical solution; thatis the solution is continuous, twice differentiable in the diffusion componentand differentiable in time. Our proof uses a probabilistic arguments andextends the results of Pham (1998) to processes with a pure jump componentwhere the jump intensity is modulated by a diffusion process. This result isparticularly useful in some applications to pricing and hedging of financialand actuarial instruments, and we provide an example to pricing of CAT bond.

Classical solutions of the Backward PIDE for Markov Modulated Marked Point Processes and Applications to CAT Bonds