  We suppose that a L\'evy process is observed at discrete time points.
Starting from an asymptotically minimax family of estimators for the continuous
part of the L\'evy Khinchine characteristics, i.e., the covariance, we derive a
data-driven parameter choice for the frequency of estimating the covariance. We
investigate a Lepski\u{i}-type stopping rule for the adaptive procedure.
Consequently, we use a balancing principle for the best possible data-driven
parameter. The adaptive estimator achieves almost the optimal rate. Numerical
experiments with the proposed selection rule are also presented.
