  The estimation of the diffusion matrix $\Sigma$ of a high-dimensional,
possibly time-changed L\'evy process is studied, based on discrete observations
of the process with a fixed distance. A low-rank condition is imposed on
$\Sigma$. Applying a spectral approach, we construct a weighted least-squares
estimator with nuclear-norm-penalisation. We prove oracle inequalities and
derive convergence rates for the diffusion matrix estimator. The convergence
rates show a surprising dependency on the rank of $\Sigma$ and are optimal in
the minimax sense for fixed dimensions. Theoretical results are illustrated by
a simulation study.
