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Asymptotic properties of power variations of Lévy processes

Published online by Cambridge University Press:  19 June 2007

Jean Jacod
Jean Jacod*
Affiliation:
Institut de mathématiques de Jussieu, 175 rue du Chevaleret 75 013 Paris, France (CNRS – UMR 7586, and Université Pierre et Marie Curie–P6); jj@ccr.jussieu.fr
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Abstract

We determine the asymptotic behavior of the realized power variations, and more generally of sums of a given function f evaluated at the increments of a Lévy process betweenthe successive times iΔn for i = 0,1,...,n. One can elucidate completely the first order behavior, that is the convergence in probability of such sums, possibly after normalization and/orcentering: it turns out that there is a rather wide variety of possible behaviors, depending on the structure of jumps and on the chosen test function f. As for the associated central limit theorem,one can show some versions of it, but unfortunately in a limited number of cases only: in some other cases a CLT just does not exist.

Keywords

Central limit theoremquadratic variationpower variationLévy processes.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 11: Special Issue: "Stochastic analysis and mathematical finance" in honor of Nicole El Karoui's 60th birthdaySpecial Issue: "Stochastic analysis and mathematical finance" in honor of Nicole El Karoui's 60th birthday , June 2007 , pp. 173 - 196
Copyright
© EDP Sciences, SMAI, 2007

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