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

Abstract

We consider random variables of the form $F=f(V_1,...,V_n)$ where $f$ is a smooth function and V_i,i\in\mathbbN are random variables with absolutely continuous law $p_i(y).$ We assume that $p_i,i=1,...,n$ are piecewise differentiable and we develop a differential calculus of Malliavin type based on \partial\lnp_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 Lévy process