Abstract

We consider the problem of estimating the density of the process associated with the small jumps of a pure jump Lévy process, possibly of infinite variation, from discrete observations of one trajectory. The interest of such a question relies on the observation that even when the Lévy measure is known, the density of the increments of the small jumps of the process cannot be computed. We discuss results both from low and high frequency observations. In a low frequency setting, assuming the Lévy density associated with the big jumps is known, a spectral estimator relying on the deconvolution structure of the problem achieves minimax parametric rates of convergence with respect to the integrated L2 loss, up to a log factor. In a high frequency setting it becomes feasible to eliminate the assumption regarding the knowledge of the Lévy measure of the large jumps. In that case the rate of convergence depends on the sampling scheme and on the behaviour of the Lévy measure in a neighborhood of zero. We show that the rate we find is minimax up to a log-factor. Additionally, we propose an adaptive penalized procedure to select the cutoff parameter. These results are extended to encompass the case where a Brownian component is present in the Lévy process. Furthermore, we illustrate our findings through an extensive simulation study. This is joint work with Céline Duval and Taher Jalal.