  Given the observation of a high-dimensional Ornstein-Uhlenbeck (OU) process
in continuous time, we proceed to the inference of the drift parameter under a
row-sparsity assumption. Towards that aim, we consider the negative
log-likelihood of the process, penalized by an $\ell^1$-penalization (Lasso and
Adaptive Lasso). We provide both non-asymptotic and asymptotic results for this
procedure, by means of a sharp oracle inequality, and a limit theorem in the
long-time asymptotics, including asymptotic consistency for variable selection.
As a by-product, we point out the fact that for the Ornstein-Uhlenbeck process,
one does not need an assumption of restricted eigenvalue type in order to
derive fast rates for the Lasso, while it is well-known to be mandatory for
linear regression for instance. Numerical results illustrate the benefits of
this penalized procedure compared to standard maximum likelihood approaches
both on simulations and real-world financial data.
