The Annals of Statistics

Estimation for Lévy processes from high frequency data within a long time interval

Fabienne Comte and Valentine Genon-Catalot

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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.

Article information

Ann. Statist. Volume 39, Number 2 (2011), 803-837.

First available in Project Euclid: 9 March 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation 62M05: Markov processes: estimation
Secondary: 60G51: Processes with independent increments; Lévy processes

Adaptive nonparametric estimation high frequency data Lévy processes projection estimators power variation


Comte, Fabienne; Genon-Catalot, Valentine. Estimation for Lévy processes from high frequency data within a long time interval. Ann. Statist. 39 (2011), no. 2, 803--837. doi:10.1214/10-AOS856.

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