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

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