Ester Mariucci est Professeur à l'Université de Versailles Saint-Quentin.
Titre : Nonparametric Density Estimation for Lévy Processes
Résumé : There is a rich literature on approximations of the small jumps of a Lévy process, with a particular focus on Gaussian approximations. These approximations are often used to simplify analysis, but their validity depends critically on the observation scheme, in particular whether the data are collected at high or low frequency. In many situations, the Gaussian approximation may not be appropriate, and a more precise understanding of the law of small jumps is necessary to investigate alternative approximations.
In this talk, I focus on the problem of estimating the density of the process associated with the small jumps of a pure jump Lévy process, possibly of infinite variation, based on discrete observations of a single trajectory. Studying this density allows us to better understand the distribution of small jumps and assess how closely they resemble Gaussian or other approximations.
We discuss results in both low-frequency and high-frequency observation settings. In the low-frequency case, assuming that the Lévy density associated with large jumps is known, a spectral estimator exploiting the deconvolution structure of the problem achieves minimax parametric rates of convergence with respect to the integrated
L^2 loss, up to a logarithmic factor. In the high-frequency case, it becomes possible to relax the assumption on the knowledge of the Lévy measure for large jumps; the rate of convergence then depends on the sampling scheme and the behavior of the Lévy measure near zero, and is minimax up to a logarithmic factor. We also propose an adaptive penalized procedure to select the cutoff parameter. These results are further extended to the case where a Brownian component is present in the Lévy process. Finally, we illustrate our findings with an extensive simulation study.
This is joint work with Céline Duval and Taher Jalal.