Minimax rates for the covariance estimation of multi-dimensional L\'evy processes with high-frequency data

This article studies nonparametric methods to estimate the co-integrated volatility for multi-dimensional L'evy 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 $1\mathrm{/}n$, the rates of convergence are $1\mathrm{/}\sqrt{n}$ if $r\le 1$ and $(n\log n{)}^{(r-2)\mathrm{/}2}$ 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.