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 of 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 Lévy processes. Given $n$ observations of increments over intervals of length $1/n$, the rates of convergence are $1/\sqrt{n}$ if $r\leq 1$ and $(n\log n)^{(r-2)/2}$ if $r>1$, where $r$ is the co-jump activity index and corresponds to the intensity of dependent jumps. These rates are optimal in a minimax sense. We bound the co-jump activity index from below by the harmonic mean of the jump activity indices of the components. Finally, we assess the efficiency of our estimator by comparing it with estimators in the existing literature.