An oracle inequality for penalised projection estimation of Lévy densities from high-frequency observations

We consider a multivariate Lévy process given by the sum of a Brownian motion with drift and an independent time-homogeneous pure jump process governed by a Lévy density.We assume that observation of a sample path takes place on an equidistant discrete time grid. Following Grenander's method of sieves, we construct families of nonparametric projection estimators for the restriction of a Lévy density to bounded sets away from the origin. Moreover, we introduce a data-driven penalisation criterion to select an estimator within a given family, where we measure the estimation error in an L ²-norm. Furthermore, we give sufficient conditions on the penalty such that an oracle inequality holds. As an application, we prove adaptiveness for sufficiently smooth Lévy densities in some Sobolev space and explicitly derive the rate of convergence.