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 $V_i,i\in \mathbb{N}$, 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 $E({\partial }_i\varphi (F)G)=E(\varphi (F)H_i(F,G))$, 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.

Published at http://dx.doi.org/10.1214/105051606000000592 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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