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

Vlad Bally; Marie-Pierre Bavouzet; Marouen Messaoud

doi:10.1214/105051606000000592

paper

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

arXiv:math/0702884 · doi:10.1214/105051606000000592

Abstract

We consider random variables of the form $F = f (V_{1}, ..., V_{n})$ , where $f$ is a smooth function and $V_{i}, i \in N$ , are random variables with absolutely continuous law $p_{i} (y) d y$ . 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 $E (\partial_{i} ϕ (F) G) = E (ϕ (F) H_{i} (F, G))$ , 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)