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We consider random variables of the form $F=f(V_1,...,V_n)$, where $f$ is asmooth function and $V_i,i\in\mathbb{N}$, are random variables with absolutelycontinuous law $p_i(y) dy$. We assume that $p_i$, $i=1,...,n$, are piecewisedifferentiable and we develop a differential calculus of Malliavin type basedon $\partial\ln p_i$. This allows us to establish an integration by partsformula $E(\partial_i\phi(F)G)=E(\phi(F)H_i(F,G))$, where $H_i(F,G)$ is arandom variable constructed using the differential operators acting on $F$ and$G.$ We use this formula in order to give numerical algorithms for sensitivitycomputations in a model driven by a L\'{e}vy process.

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