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 $d=2$. We consider a class of jump diffusion processes whose invariant density belongs to some H\"older 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.