Nonparametric inference for discretely sampled Lévy processes

Abstract

Given a sample from a discretely observed Lévy process $X=(X_t{)}_{t\ge 0}$ of the finite jump activity, the problem of nonparametric estimation of the Lévy density $\rho$ corresponding to the process $X$ is studied. An estimator of $\rho$ is proposed that is based on a suitable inversion of the Lévy-Khintchine formula and a plug-in device. The main results of the paper deal with upper risk bounds for estimation of $\rho$ over suitable classes of Lévy triplets. The corresponding lower bounds are also discussed.

38 pages

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• Nonparametric estimation for Lévy processes from low-frequency observations
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• Nonparametric inference on Lévy measures of compound Poisson-driven Ornstein-Uhlenbeck processes under macroscopic discrete observations
• On a linear functional for infinitely divisible moving average random fields
• Joint estimation for volatility and drift parameters of ergodic jump diffusion processes via contrast function