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

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

2007-02-28

arxiv.org/abs/math/0702884v1

Annals of Applied Probability 2007, Vol. 17, No. 1, 33-66

Mathematics

Probability

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

Scientific paper

10.1214/105051606000000592

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.

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