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