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 hypothesis 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。本文旨在研究中间制度下不变密度估计的收敛速率，假设在$T$趋于无穷时，可获得过程$(X_t)_{t \in [0, T]}$的离散观测。我们给出了所提出核密度估计量的收敛速率，以及离散化步长$\Delta_n$的条件，该步长作为中间制度与连续情形之间的阈值。在中间制度下，收敛速率为$n^{- \frac{2 \beta}{2 \beta + 1}}$，其中$\beta$为不变密度的光滑性。随后，我们通过所有可能估计量集合上的下界补充了先前发现的上界，该下界提供了相同的收敛速率：这意味着无法提出实现更优收敛速率的不同估计量。这一结论通过双假设方法获得，其最具挑战性的部分在于界定两个模型律之间的Hellinger距离。关键点在于得分函数的Malliavin表示，该表示使我们能够通过依赖于Malliavin权重的量来界定Hellinger距离。