Asymptotic properties of power variations of Lévy processes

Jean Jacod

ESAIM: Probability and Statistics (2007)

  • Volume: 11, page 173-196
  • ISSN: 1292-8100

Abstract

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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.

How to cite

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Jacod, Jean. "Asymptotic properties of power variations of Lévy processes." ESAIM: Probability and Statistics 11 (2007): 173-196. <http://eudml.org/doc/250133>.

@article{Jacod2007,
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 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. },
author = {Jacod, Jean},
journal = {ESAIM: Probability and Statistics},
keywords = {Central limit theorem; quadratic variation; power variation; Lévy processes.; central limit theorem; Lévy processes},
language = {eng},
month = {6},
pages = {173-196},
publisher = {EDP Sciences},
title = {Asymptotic properties of power variations of Lévy processes},
url = {http://eudml.org/doc/250133},
volume = {11},
year = {2007},
}

TY - JOUR
AU - Jacod, Jean
TI - Asymptotic properties of power variations of Lévy processes
JO - ESAIM: Probability and Statistics
DA - 2007/6//
PB - EDP Sciences
VL - 11
SP - 173
EP - 196
AB - 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.
LA - eng
KW - Central limit theorem; quadratic variation; power variation; Lévy processes.; central limit theorem; Lévy processes
UR - http://eudml.org/doc/250133
ER -

References

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  1. Y. Aït-Sahalia and J. Jacod, Volatility estimators for discretely sampled Lévy processes. To appear in Annals of Statistics (2005).  
  2. 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).  
  3. O.E. Barndorff-Nielsen and N. Shephard, Realised power variation and stochastic volatility. Bernoulli 9 (2003) 243–265. Correction published in pages 1109–1111.  
  4. 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.  
  5. O.E. Barndorff-Nielsen, N. Shephard and M. Winkel, Limit theorems for multipower variation in the presence of jumps. Stoch. Processes Appl.116 (2006) 796–806.  
  6. A.N. Borodin and I.A. Ibragimov, Limit Theorems for Functionals of Random Walks. Proceedings Staklov Inst. Math. 195, A.M.S. (1995).  
  7. J. Jacod and A. Shiryaev, Limit Theorems for Stochastic Processes. 2nd ed., Springer-Verlag, Berlin (2003).  
  8. J. Jacod and P. Protter, Asymptotic error distributions for the Euler method for stochastic differential equations. Ann. Probab.26 (1998) 267–307.  
  9. J. Jacod, The Euler scheme for Lévy driven stochastic differential equations: limit theorems. Ann. Probab.32 (2004) 1830–1972.  
  10. J. Jacod, A. Jakubowski and J. Mémin, On asymptotic error in discretization of processes. Ann. Probab.31 (2003) 592–608.  
  11. D. Lépingle, La variation d'ordre p des semimartingales. Z. für Wahr. Th.36 (1976) 285–316.  
  12. C. Mancini, Disentangling the jumps of the diffusion in a geometric jumping Brownian motion. Giornale dell'Instituto Italiano degli AttuariLXIV (2001) 19–47.  
  13. 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.  

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