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.

中文翻译：

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Malliavin 演算用于最佳估计中间状态下离散观察到的扩散的不变密度

令$(X_t)_{t \ge 0}$ 为一维随机微分方程的解。我们的目的是研究在中间状态下估计不变密度的收敛速度，假设当倾向于$\infty$。我们找到了与我们提出的核密度估计器相关的收敛速度，以及离散化步骤 $\Delta_n$ 的条件，该条件在中间状态和连续情况之间起到阈值的作用。在中间状态下，收敛速度为 $n^{- \frac{2 \beta}{2 \beta + 1}}$，其中 $\beta$ 是不变密度的平滑度。之后，我们用所有可能的估计器集合的下限来补充先前找到的上限，这提供了相同的收敛速度：这意味着不可能提出一个不同的估计器来实现更好的收敛速度。这是通过二假设法得到的；最具挑战性的部分在于限定两个模型的定律之间的 Hellinger 距离。关键点是分数函数的 Malliavin 表示，它允许我们通过取决于 Malliavin 权重的量来限制 Hellinger 距离。