Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Optimal estimation of local time and occupation time measure for an α-stable Levy process

Chiara Amorino, Arturo Jaramillo, Mark Podolskij

Abstract

We present a novel theoretical result on estimation of local time and occupation time measure of an α-stable Lévy process with α in (1, 2). Our approach is based upon computing the conditional expectation of the desired quantities given high frequency data, which is an L^2-optimal statistic by construction. We prove the corresponding stable central limit theorems and discuss a statistical application. In particular, this work extends the results of [Ivanovs and i Podolskij (2021)], which investigated the case of the Brownian motion.

Optimal estimation of local time and occupation time measure for an α-stable Levy process

Abstract

We present a novel theoretical result on estimation of local time and occupation time measure of an α-stable Lévy process with α in (1, 2). Our approach is based upon computing the conditional expectation of the desired quantities given high frequency data, which is an L^2-optimal statistic by construction. We prove the corresponding stable central limit theorems and discuss a statistical application. In particular, this work extends the results of [Ivanovs and i Podolskij (2021)], which investigated the case of the Brownian motion.
Paper Structure (14 sections, 7 theorems, 89 equations)

This paper contains 14 sections, 7 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. The setting and overview
  3. Related literature
  4. Outline of the paper
  5. Preliminaries and main results
  6. Local times and stable convergence
  7. Main results
  8. Proof of main results
  9. Proof of Theorem \ref{['th: main local']}
  10. Proof of the auxiliary results
  11. Proof of Lemma \ref{['l:lemmaFourierLT']}
  12. Proof of Proposition \ref{['l: estim local']}
  13. Proof of Lemma \ref{['l: moments L and O']}
  14. Proof of Lemmas \ref{['l: edge local']} and \ref{['l: edge occ']}

Key Result

Theorem 1.1

Suppose that $\phi:\mathbb{R}\rightarrow \mathbb{R}$ is a function satisfying $\phi,\phi^2\in L^1(\mathbb{R};\mathbb{R})$. Then, for every $t>0$,

Theorems & Definitions (15)

  • Theorem 1.1: Theorem 4 in J04
  • Lemma 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 5 more