Nonparametric inference on L\'evy measures and copulas

TL;DR

This paper introduces nonparametric methods for estimating multivariate Lévy measures and copulas from high-frequency data, providing theoretical convergence results, extensions to irregular sampling, and practical applications.

Contribution

It develops new nonparametric estimators for Lévy measures and Pareto-Lévy copulas, with proven weak convergence and novel analysis of their properties under various sampling schemes.

Findings

Estimators achieve a convergence rate of $k_n^{-1/2}$.

Analytic properties of Pareto-Lévy copulas are characterized.

Simulation and real data applications demonstrate estimator performance.

Abstract

In this paper nonparametric methods to assess the multivariate L\'{e}vy measure are introduced. Starting from high-frequency observations of a L\'{e}vy process $\mathbf{X}$, we construct estimators for its tail integrals and the Pareto-L\'{e}vy copula and prove weak convergence of these estimators in certain function spaces. Given n observations of increments over intervals of length $\Delta_n$, the rate of convergence is $k_n^{-1/2}$ for $k_n=n\Delta_n$ which is natural concerning inference on the L\'{e}vy measure. Besides extensions to nonequidistant sampling schemes analytic properties of the Pareto-L\'{e}vy 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.