Let $(X_t)_{t \ge 0}$ be solution of a one-dimensional stochastic differential equation. Our aim is to study the convergence rate for the estimation of the invariant density in intermediate regime, assuming that a discrete observation of the process $(X_t)_{t \in [0, T]}$ is available, when $T$ tends to $\infty$. We find the convergence rates associated to the kernel density estimator we proposed and a condition on the discretization step $\Delta_n$ which plays the role of threshold between the intermediate regime and the continuous case. In intermediate regime the convergence rate is $n^{- \frac{2 \beta}{2 \beta + 1}}$, where $\beta$ is the smoothness of the invariant density. After that, we complement the upper bounds previously found with a lower bound over the set of all the possible estimator, which provides the same convergence rate: it means it is not possible to propose a different estimator which achieves better convergence rates. This is obtained by the two hypotheses method; the most challenging part consists in bounding the Hellinger distance between the laws of the two models. The key point is a Malliavin representation for a score function, which allows us to bound the Hellinger distance through a quantity depending on the Malliavin weight.

翻译：设$(X_t)_{t \ge 0}$为一维随机微分方程的解。本文旨在研究当$T$趋于无穷时，基于过程$(X_t)_{t \in [0, T]}$的离散观测，在中间制度下估计不变密度的收敛速度。我们获得了所提出的核密度估计量的收敛速度，以及离散步长$\Delta_n$作为中间制度与连续情形之间阈值的条件。在中间制度下，收敛速度为$n^{- \frac{2 \beta}{2 \beta + 1}}$，其中$\beta$为不变密度的光滑度。随后，我们通过所有可能估计量集合上的下界对先前得到的上界进行补充，该下界给出了相同的收敛速度：这意味着不可能提出实现更优收敛速度的不同估计量。这一结果通过双假设方法获得，其中最困难的部分在于界定两种模型定律之间的Hellinger距离。关键在于得分函数的Malliavin表示，它使我们能够通过依赖于Malliavin权重的量来界定Hellinger距离。