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

Chiara Amorino, Arnaud Gloter

TL;DR

The 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 $\infty$.

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 $Δ_n$ 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.

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

TL;DR

The 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 T\infty$.

Abstract

Let 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 is available, when tends to . We find the convergence rates associated to the kernel density estimator we proposed and a condition on the discretization step which plays the role of threshold between the intermediate regime and the continuous case. In intermediate regime the convergence rate is , where 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.
Paper Structure (26 sections, 20 theorems, 250 equations)

This paper contains 26 sections, 20 theorems, 250 equations.

Table of Contents

  1. Introduction
  2. Model Assumptions
  3. Construction estimator and main results
  4. Proofs
  5. Proof of Proposition \ref{['prop: bound var discrete d=1']}
  6. Proof of Theorem \ref{['th: discrete d =1']}
  7. Proof of Theorem \ref{['th: borne inf discrete']}
  8. Proof of Proposition \ref{['prop: control Fisher sans Malliavin']}
  9. Proof of Proposition \ref{['prop: bound Malliavin']}
  10. proof of technical results
  11. Proof of Lemma \ref{['l: hyper contract']}
  12. Proof of Lemma \ref{['l: AB']}
  13. Proof of Lemma \ref{['L: maj grossiere Malliavin']}
  14. Proof of Lemma \ref{['l: bound stationarity']}
  15. Proof of Proposition \ref{['p: bound W1']}
  16. ...and 11 more sections

Key Result

Theorem 1

Under A1-A2, for any $\tau > 0$, $(s, t) \in [0, \infty)^2$, $0 < t-s < \tau$, the unique weak solution of eq: model admits a transition density $p_{t - s}(x,y)$ which is continuous in $x, y \in \mathbb{R}$. Moreover, there exist $\lambda_0 \in (0, 1]$ and ${C_0}, c_0 \ge 1$ such that, for any $(s,t and the constants depend only on $\tau ,~a_\text{min},~b_0,~a_0,~ a_1$ and $b_1$. Moreover, for $k=

Theorems & Definitions (42)

  • Definition 1
  • Definition 2
  • Theorem 1: MenPes
  • Proposition 1
  • Theorem 2
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 3
  • Remark 4
  • ...and 32 more