Malliavin calculus for the optimal estimation of the invariant density of discretely observed diffusions in intermediate regime
Abstract
Let $(X_t)_{t \ge 0}$ be solution of a onedimensional 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.
 Publication:

arXiv eprints
 Pub Date:
 August 2022
 DOI:
 10.48550/arXiv.2208.03253
 arXiv:
 arXiv:2208.03253
 Bibcode:
 2022arXiv220803253A
 Keywords:

 Mathematics  Statistics Theory