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