Spectral-free estimation of Lévy densities in high-frequency regime

Abstract

We construct an estimator of the Lévy density of a pure jump Lévy process, possibly of infinite variation, from the discrete observation of one trajectory at high frequency. The novelty of our procedure is that we directly estimate the Lévy density relying on a pathwise strategy, whereas existing procedures rely on spectral techniques. By taking advantage of a compound Poisson approximation, we circumvent the use of spectral techniques and in particular of the Lévy–Khintchine formula. A linear wavelet estimator is built and its performance is studied in terms of ${\mathit{L}}_{\mathit{p}}$ loss functions, $\mathit{p}\ge 1$, over Besov balls. We recover classical nonparametric rates for finite variation Lévy processes and for a nonparametric class of symmetric infinite variation Lévy processes. We show that the procedure is robust when the estimation set gets close to the critical value 0 and also discuss its robustness to the presence of a Brownian part.