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

We consider random variables of the form F = f(V-1,..., V-n), where f is a smooth function and Vi, i EN, are random variables with absolutely continuous law p(i) (y) dy. We assume that p(i), i = 1,..., n, are piecewise differentiable and we develop a differential calculus of Malliavin type based on a In pi. This allows us to establish an integration by parts formula E(delta i phi(F)G) = E(phi(F)H-i(F, G)), where H-i(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 Levy process.