Estimation for L\'{e}vy processes from high frequency data within a long time interval

TL;DR

This paper develops an adaptive nonparametric Fourier-based estimator for the Lévy density of Lévy processes using high-frequency data over long time intervals, providing theoretical risk bounds and simulation validation.

Contribution

It introduces a novel Fourier-based adaptive estimator for Lévy densities from high-frequency data, with theoretical risk bounds and analysis of convergence rates.

Findings

Estimator achieves optimal convergence rates.

Provides bounds for ${\ m L}^2$-risk.

Simulation results confirm theoretical properties.

Abstract

In this paper, we study nonparametric estimation of the L\'{e}vy density for L\'{e}vy processes, with and without Brownian component. For this, 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. We use a Fourier approach to construct an adaptive nonparametric estimator of the L\'{e}vy 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.