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.

翻译：我们的目标是估算与低维跳跃的随机差异方程式相关的不变化密度,即1美元=1美元和2美元=2美元。我们考虑的跳跃扩散过程类别,其不变化密度属于某种H\"older 空间。首先,在层面一,我们显示内核密度估计值达到合率$\frac{1 ⁇ T}$,这是没有跳跃时的最佳率。这改善了[阿莫里诺,格洛特(2021年)]获得的汇合率,这取决于Blumenthal-Getoor指数$=1美元,等于$\frac\log T ⁇ T}$=2美元。第二,我们表明不可能找到一个估算率更快的估算器。事实上,我们得到一些与相同比率($üfrac{1 ⁇ T},\fracT ⁇ T ⁇ $(2021年)相同的较低界限。这取决于Blumenthal-Getoor指数。最后,我们在一个案例中获取了标准的正常度。