We consider the solution X = (Xt) t$\ge$0 of a multivariate stochastic differential equation with Levy-type jumps and with unique invariant probability measure with density $\mu$. We assume that a continuous record of observations X T = (Xt) 0$\le$t$\le$T is available. In the case without jumps, Reiss and Dalalyan (2007) and Strauch (2018) have found convergence rates of invariant density estimators, under respectively isotropic and anisotropic H{o}lder smoothness constraints, which are considerably faster than those known from standard multivariate density estimation. We extend the previous works by obtaining, in presence of jumps, some estimators which have the same convergence rates they had in the case without jumps for d $\ge$ 2 and a rate which depends on the degree of the jumps in the one-dimensional setting. We propose moreover a data driven bandwidth selection procedure based on the Goldensh-luger and Lepski (2011) method which leads us to an adaptive non-parametric kernel estimator of the stationary density $\mu$ of the jump diffusion X. Adaptive bandwidth selection, anisotropic density estimation, ergodic diffusion with jumps, L{e}vy driven SDE

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各向异性类遍历跳跃扩散过程的不变密度自适应估计

我们考虑具有 Levy 类型跳跃和具有密度 $\mu$ 的唯一不变概率测度的多元随机微分方程的解 X = (Xt) t$\ge$0。我们假设连续观测记录 XT = (Xt) 0$\le$t$\le$T 是可用的。在没有跳跃的情况下，Reiss 和 Dalalyan (2007) 以及 Strauch (2018) 分别发现了在各向同性和各向异性 H{o}lder 平滑约束下的不变密度估计量的收敛速度，这比从标准多元密度已知的要快得多估计。我们通过在存在跳跃的情况下获得一些估计量来扩展之前的工作，这些估计量具有与 d$\ge$ 2 没有跳跃的情况下相同的收敛率，以及取决于跳跃程度的速率 -维度设置。