In this paper we consider an ergodic diffusion process with jumps whose drift coefficient depends on µ and volatility coefficient depends on σ, two unknown parameters. We suppose that the process is discretely observed at the instants (t n i )i=0,...,n with ∆n = sup i=0,...,n−1 (t n i+1 − t n i ) → 0. We introduce an estimator of θ := (µ, σ), based on a contrast function, which is asymptotically gaussian without requiring any conditions on the rate at which ∆n → 0, assuming a finite jump activity. This extends earlier results where a condition on the step discretization was needed (see [13], [28]) or where only the estimation of the drift parameter was considered (see [2]). In general situations, our contrast function is not explicit and in practise one has to resort to some approximation. We propose explicit approximations of the contrast function, such that the estimation of θ is feasible under the condition that n∆ k n → 0 where k > 0 can be arbitrarily large. This extends the results obtained by Kessler [17] in the case of continuous processes.Efficient drift estimation, efficient volatility estimation,ergodic properties, high frequency data, Lévydriven SDE, thresholding methods.

We consider the solution X = (Xt) t≥0 of a multivariate stochastic differential equation with Levy-type jumps and with unique invariant probability measure with density µ. We assume that a continuous record of observations X T = (Xt) 0≤t≤T is available. In the case without jumps, Reiss and Dalalyan [7] and Strauch [24] have found convergence rates of invariant density estimators, under respectively isotropic and anisotropic Hölder smoothness constraints, which are considerably faster than those known from standard multivariate density estimation. We extend the previous works by obtaining, in presence of jumps, some estimators which have the same convergence rates they had in the case without jumps for d ≥ 2 and a rate which depends on the degree of the jumps in the one-dimensional setting. We propose moreover a data driven bandwidth selection procedure based on the Goldenshluger and Lepski method [11] which leads us to an adaptive non-parametric kernel estimator of the stationary density µ of the jump diffusion X.