Minimax rates for the covariance estimation of multi-dimensional Lévy processes with high-frequency data

Abstract

This article studies nonparametric methods to estimate the co-integrated volatility for multi-dimensional Lévy processes with high frequency data. We construct a spectral estimator for the co-integrated volatility and prove minimax rates for an appropriate bounded nonparametric class of semimartingales. Given $n$ observations of increments over intervals of length $1/n$, the rates of convergence are $1/\sqrt{n}$ if $r\le 1$ and $(n\log n{)}^{(r-2)/2}$ if $r>1$, which are optimal in a minimax sense. We bound the co-jump index activity from below with the harmonic mean. Finally, we assess the efficiency of our estimator by comparing it with estimators in the existing literature.

Important changes have been made in this version. This version supersedes version 1