  We consider random variables of the form $F=f(V_1,...,V_n)$, where $f$ is a
smooth function and $V_i,i\in\mathbb{N}$, 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 $\partial\ln p_i$. This allows us to establish an integration by parts
formula $E(\partial_i\phi(F)G)=E(\phi(F)H_i(F,G))$, 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 L\'{e}vy process.
