  In this paper we consider an ergodic diffusion process with jumps whose drift
coefficient depends on an unknown parameter $\theta$. We suppose that the
process is discretely observed at the instants (t n i)i=0,...,n with $\Delta$n
= sup i=0,...,n--1 (t n i+1 -- t n i) $\rightarrow$ 0. We introduce an
estimator of $\theta$, based on a contrast function, which is efficient without
requiring any conditions on the rate at which $\Delta$n $\rightarrow$ 0, and
where we allow the observed process to have non summable jumps. This extends
earlier results where the condition n$\Delta$ 3 n $\rightarrow$ 0 was needed
(see [10],[24]) and where the process was supposed to have summable jumps.
Moreover, in the case of a finite jump activity, we propose explicit
approximations of the contrast function, such that the efficient estimation of
$\theta$ is feasible under the condition that n$\Delta$ k n $\rightarrow$ 0
where k > 0 can be arbitrarily large. This extends the results obtained by
Kessler [15] in the case of continuous processes. L{\'e}vy-driven SDE,
efficient drift estimation, high frequency data, ergodic properties,
thresholding methods.
