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

TL;DR

This paper investigates the convergence rates of a kernel density estimator for the invariant density of discretely observed diffusions in the intermediate regime, establishing both upper and lower bounds and highlighting the role of Malliavin calculus.

Contribution

It introduces a new estimator for the invariant density in the intermediate regime and proves its optimal convergence rate using Malliavin calculus techniques.

Findings

Convergence rate in intermediate regime is n^{-2β/(2β+1)}.

Upper and lower bounds match, confirming optimality.

Mallivain representation aids in bounding Hellinger distance.

Abstract

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.