Optimal estimation of local time and occupation time measure for an {\alpha}-stable Levy process

TL;DR

This paper develops a new theoretical method for estimating local time and occupation time measures of an {}-stable Levy process using high-frequency data, extending previous Brownian motion results.

Contribution

It introduces an L^2-optimal estimator for local and occupation times of {}-stable Levy processes and proves stable CLTs, expanding prior work on Brownian motion.

Findings

Derived stable central limit theorems for the estimators

Extended estimation techniques from Brownian motion to {}-stable Levy processes

Provided a statistical framework for high-frequency data analysis

Abstract

We present a novel theoretical result on estimation of local time and occupation time measure of an {\alpha}-stable L\'evy process with {\alpha} 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.