Volume 11, February 2007Special Issue: "Stochastic analysis and mathematical finance" in honor of Nicole El Karoui's 60th birthday
|Page(s)||173 - 196|
|Published online||19 June 2007|
Asymptotic properties of power variations of Lévy processes
Institut de mathématiques de Jussieu, 175 rue du Chevaleret 75 013 Paris, France (CNRS – UMR 7586, and Université Pierre et Marie Curie–P6); email@example.com
Revised: 4 April 2006
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 between the 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/or centering: 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.
Mathematics Subject Classification: 60F17 / 60G51
Key words: Central limit theorem / quadratic variation / power variation / Lévy processes.
© EDP Sciences, SMAI, 2007
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