# Nonparametric estimation for pure jump Lévy processes based on high frequency data.

Abstract : In this paper, we study nonparametric estimation of the Lévy density for pure jump Lévy processes. We consider $n$ discrete time observations with step $\Delta$. The asymptotic framework is: $n$ tends to infinity, $\Delta=\Delta_n$ tends to zero while $n\Delta_n$ tends to infinity. First, we use a Fourier approach (frequency domain"): this allows to construct an adaptive nonparametric estimator and to provide a bound for the global ${\mathbb L}^2$-risk. Second, we use a direct approach (time domain") which allows to construct an estimator on a given compact interval. We provide a bound for ${\mathbb L}^2$-risk restricted to the compact interval. We discuss rates of convergence and give examples and simulation results for processes fitting in our framework.
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https://hal.archives-ouvertes.fr/hal-00358184
Contributor : Fabienne Comte <>
Submitted on : Tuesday, February 3, 2009 - 9:22:27 AM
Last modification on : Friday, September 20, 2019 - 4:34:02 PM
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Fabienne Comte, Valentine Genon-Catalot. Nonparametric estimation for pure jump Lévy processes based on high frequency data.. Stochastic Processes and their Applications, Elsevier, 2009, 119 (12), pp.4088-4123. ⟨10.1016/j.spa.2009.09.013⟩. ⟨hal-00358184⟩

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