Hostname: page-component-5b777bbd6c-gtgcz Total loading time: 0 Render date: 2025-06-24T05:19:00.987Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Y. Aït-Sahalia and J. Jacod, Volatility estimators for discretely sampled Lévy processes. To appear in Annals of Statistics (2005).
T.G. Andersen, T. Bollerslev and F.X. Diebold, Parametric and nonparametric measurement of volatility, in Handbook of Financial Econometrics, Y. Aït-Sahalia and L.P. Hansen Eds., Amsterdam: North Holland. Forthcoming (2005).
Barndorff-Nielsen, O.E. and Shephard, N., Realised power variation and stochastic volatility. Bernoulli 9 (2003) 243–265. Correction published in pages 1109–1111. CrossRef
O.E. Barndorff-Nielsen, S.E. Graversen, J. Jacod, M. Podolskij and N. Shephard, A central limit theorem for realised bipower variations of continuous semimartingales, in From Stochastic calculus to mathematical finance, the Shiryaev Festschrift, Y. Kabanov, R. Liptser, J. Stoyanov Eds., Springer-Verlag, Berlin (2006) 33–69.
Barndorff-Nielsen, O.E., Shephard, N. and Winkel, M., Limit theorems for multipower variation in the presence of jumps. Stoch. Processes Appl. 116 (2006) 796–806. CrossRef
A.N. Borodin and I.A. Ibragimov, Limit Theorems for Functionals of Random Walks. Proceedings Staklov Inst. Math. 195, A.M.S. (1995).
J. Jacod and A. Shiryaev, Limit Theorems for Stochastic Processes. 2nd ed., Springer-Verlag, Berlin (2003).
Jacod, J. and Protter, P., Asymptotic error distributions for the Euler method for stochastic differential equations. Ann. Probab. 26 (1998) 267–307.
Jacod, J., The Euler scheme for Lévy driven stochastic differential equations: limit theorems. Ann. Probab. 32 (2004) 1830–1972. CrossRef
Jacod, J., Jakubowski, A. and Mémin, J., On asymptotic error in discretization of processes. Ann. Probab. 31 (2003) 592–608.
Lépingle, D., La variation d'ordre p des semimartingales. Z. für Wahr. Th. 36 (1976) 285–316.
C. Mancini, Disentangling the jumps of the diffusion in a geometric jumping Brownian motion. Giornale dell'Instituto Italiano degli Attuari LXIV (2001) 19–47.
J. Woerner, Power and multipower variation: inference for high frequency data, in Stochastic Finance, A.N. Shiryaev, M. do Rosário Grosshino, P. Oliviera, M. Esquivel Eds., Springer-Verlag, Berlin (2006) 343–354.