  The aim of this paper is to introduce an adaptive penalized estimator for
identifying the true reduced parametric model under the sparsity assumption. In
particular, we deal with the framework where the unpenalized estimator of the
structural parameters needs simultaneously multiple rates of convergence (i.e.
the so-called mixed-rates asymptotic behavior). We introduce a Bridge-type
estimator by taking into account penalty functions involving $\ell^q$ norms
$(0<q\leq 1)$. We prove that the proposed regularized estimator satisfies the
oracle properties.
  Our approach is useful for the estimation of stochastic differential
equations in the parametric sparse setting. More precisely, under the high
frequency observation scheme, we apply our methodology to an ergodic diffusion
and introduce a procedure for the selection of the tuning parameters.
Furthermore, the paper contains a simulation study as well as a real data
prediction in order to assess about the performance of the proposed Bridge
estimator.
