Optimal convergence rates for the invariant density estimation of jump-diffusion processes

We aim at estimating the invariant density associated to a stochastic differential equation with jumps in low dimension, which is for d = 1 and 2. consider class of fully non-linear jump diffusion processes whose belongs some Hölder space. Firstly, dimension one, we show that kernel estimator achieves convergence rate 1/ T , optimal absence jumps. This improves obtained Amorino Gloter [ J. Stat. Plann. Inference 213 (2021) 106–129], depends on Blumenthal-Getoor index equal (log )/ Secondly, when coefficients are constant finite, not possible find an faster rates estimation. Indeed, get lower bounds same {1/ } mono bi-dimensional cases, respectively. Finally, obtain asymptotic normality one-dimensional case process.