Integration by parts formula for locally smooth laws and applications to sensitivity computations

We consider random variables of the form F=f(V1,..,.Vn), where f is a smooth function and Vi, i.., are random variables with absolutely continuous law pi(y).dy. We assume that pi, i=1,..,.n, are piecewise differentiable and we develop a differential calculus of Malliavin type based on .lnpi. This allows us to establish an integration by parts formula E(.i.(F)G)=E(.(F)Hi(F,.G)), where Hi(F,.G) is a random variable constructed using the differential operators acting on F and G. We use this formula in order to give numerical algorithms for sensitivity computations in a model driven by a Lévy process.