Computation of Greeks using Malliavin's calculus in jump type market models

We use the Malliavin calculus for Poisson processes in order to compute sensitivities for European options with underlying following a jump type diffusion. The main point is to settle an integration by parts formula (similar to the one in the Malliavin calculus) for a general multidimensional random variable which has an absolutely continuous law with differentiable density. We give an explicit expression of the differential operators involved in this formula and this permits to simulate them and consequently to run a Monte Carlo algorithm