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April 2011 Estimation for Lévy processes from high frequency data within a long time interval
Fabienne Comte, Valentine Genon-Catalot
Ann. Statist. 39(2): 803-837 (April 2011). DOI: 10.1214/10-AOS856


In this paper, we study nonparametric estimation of the Lévy density for Lévy processes, with and without Brownian component. For this, we consider n discrete time observations with step Δ. The asymptotic framework is: n tends to infinity, Δ=Δn tends to zero while nΔn tends to infinity. We use a Fourier approach to construct an adaptive nonparametric estimator of the Lévy density and to provide a bound for the global ${\mathbb{L}}^{2}$-risk. Estimators of the drift and of the variance of the Gaussian component are also studied. We discuss rates of convergence and give examples and simulation results for processes fitting in our framework.


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Fabienne Comte. Valentine Genon-Catalot. "Estimation for Lévy processes from high frequency data within a long time interval." Ann. Statist. 39 (2) 803 - 837, April 2011.


Published: April 2011
First available in Project Euclid: 9 March 2011

zbMATH: 1215.62084
MathSciNet: MR2816339
Digital Object Identifier: 10.1214/10-AOS856

Primary: 62G05 , 62M05
Secondary: 60G51

Keywords: adaptive nonparametric estimation , High frequency data , Lévy processes , power variation , projection estimators

Rights: Copyright © 2011 Institute of Mathematical Statistics


Vol.39 • No. 2 • April 2011
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