Malliavin calculus for the optimal estimation of the invariant density of discretely observed diffusions in intermediate regime
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arXiv
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| Natura: | Preprint |
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2022
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| _version_ | 1866914697835446272 |
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| author | Amorino, Chiara Gloter, Arnaud |
| author_facet | Amorino, Chiara Gloter, Arnaud |
| contents | 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 $Δ_n$ which plays the role of threshold between the intermediate regime and the continuous case. In intermediate regime the convergence rate is $n^{- \frac{2 β}{2 β+ 1}}$, where $β$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_03253 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Malliavin calculus for the optimal estimation of the invariant density of discretely observed diffusions in intermediate regime Amorino, Chiara Gloter, Arnaud Statistics Theory 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 $Δ_n$ which plays the role of threshold between the intermediate regime and the continuous case. In intermediate regime the convergence rate is $n^{- \frac{2 β}{2 β+ 1}}$, where $β$ 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. |
| title | Malliavin calculus for the optimal estimation of the invariant density of discretely observed diffusions in intermediate regime |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2208.03253 |