  We investigate the problem of estimating the drift parameter of a
high-dimensional L\'evy-driven Ornstein--Uhlenbeck process under sparsity
constraints. It is shown that both Lasso and Slope estimators achieve the
minimax optimal rate of convergence (up to numerical constants), for tuning
parameters chosen independently of the confidence level, which improves the
previously obtained results for standard Ornstein--Uhlenbeck processes. The
results are nonasymptotic and hold both in probability and conditional
expectation with respect to an event resembling the restricted eigenvalue
condition.
