Nonparametric inference on Lévy measures of compound Poisson-driven Ornstein-Uhlenbeck processes under macroscopic discrete observations

Abstract

This study examines a nonparametric inference on a stationary Lévy-driven Ornstein-Uhlenbeck (OU) process $X=(X_{t})_{t\geq 0}$ with a compound Poisson subordinator. We propose a new spectral estimator for the Lévy measure of the Lévy-driven OU process $X$ under macroscopic observations. We also derive, for the estimator, multivariate central limit theorems over a finite number of design points, and high-dimensional central limit theorems in the case wherein the number of design points increases with an increase in the sample size. Built on these asymptotic results, we develop methods to construct confidence bands for the Lévy measure and propose a practical method for bandwidth selection.