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

Abstract

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 ${(X_t)}_{0\le t\le T}$, when the dimension d is such that $d\ge 3$. From the continuous observation of the sampling path on $[0,T]$ we show that, under anisotropic Hölder smoothness constraints, kernel based estimators can achieve fast convergence rates. In particular, they are as fast as the ones found by Dalalyan and Reiss [11] for the estimation of the invariant density in the case without jumps under isotropic Hölder smoothness constraints. Moreover, they are faster than the ones found by Strauch [32] for the invariant density estimation of continuous stochastic differential equations, under anisotropic Hölder 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.