Rate of estimation for the stationary distribution of jump-processes over anisotropic Holder classes.

We study the problem of the non-parametric estimation for the density π of the stationary distribution of the multivariate stochastic differential equation with jumps (Xt)0≤t≤T , when the dimension d is such that d ≥ 3. From the continuous observa- tion of the sampling path on [0, T ] we show that, under anisotropic H ̈older smoothness constraints, kernel based estimators can achieve fast convergence rates. In particu- lar, they are as fast as the ones found by Dalalyan and Reiss [9] for the estimation of the invariant density in the case without jumps under isotropic H ̈older smooth- ness constraints. Moreover, they are faster than the ones found by Strauch [29] for the invariant density estimation of continuous stochastic differential equations, under anisotropic H ̈older smoothness constraints. Furthermore, we obtain a minimax lower bound on the L2-risk for pointwise estimation, with the same rate up to a log(T) term. It implies that, on a class of diffusions whose invariant density belongs to the anisotropic Holder class we are considering, it is impossible to find an estimator with a rate of estimation faster than the one we propose.