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Let $(X_t)_{t \ge 0}$ be solution of a one-dimensional stochasticdifferential equation. Our aim is to study the convergence rate for theestimation of the invariant density in intermediate regime, assuming that adiscrete 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 kerneldensity estimator we proposed and a condition on the discretization step$\Delta_n$ which plays the role of threshold between the intermediate regimeand the continuous case. In intermediate regime the convergence rate is $n^{-\frac{2 \beta}{2 \beta + 1}}$, where $\beta$ is the smoothness of the invariantdensity. After that, we complement the upper bounds previously found with alower bound over the set of all the possible estimator, which provides the sameconvergence rate: it means it is not possible to propose a different estimatorwhich achieves better convergence rates. This is obtained by the two hypothesismethod; the most challenging part consists in bounding the Hellinger distancebetween the laws of the two models. The key point is a Malliavin representationfor a score function, which allows us to bound the Hellinger distance through aquantity depending on the Malliavin weight.

Malliavin calculus for the optimal estimation of the invariant density of discretely observed diffusions in intermediate regime