Nonparametric inference on Lévy measures and copulas

Abstract

In this paper nonparametric methods to assess the multivariate Lévy measure are introduced. Starting from high-frequency observations of a Lévy process X, we construct estimators for its tail integrals and the Pareto-Lévy copula and prove weak convergence of these estimators in certain function spaces. Given n observations of increments over intervals of length δn, the rate of convergence is k-1/2n for kn = nδn which is natural concerning inference on the Lévy measure. Besides extensions to nonequidistant sampling schemes analytic properties of the Pareto-Lévy copula which, to the best of our knowledge, have not been mentioned before in the literature are provided as well.We conclude with a short simulation study on the performance of our estimators and apply them to real data.