Optimal estimation of the local time and the occupation time measure for an α -stable Lévy process
Volume 11, Issue 2 (2024), pp. 149–168
Pub. online: 6 February 2024
Type: Research Article
Open Access
Open Access
Received
6 May 2023
6 May 2023
Revised
31 October 2023
31 October 2023
Accepted
2 January 2024
2 January 2024
Published
6 February 2024
6 February 2024
Abstract
A novel theoretical result on estimation of the local time and the occupation time measure of an α-stable Lévy process with $\alpha \in (1,2)$ is presented. The 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. The corresponding stable central limit theorems are proved and a statistical application is discussed. In particular, this work extends the results of [20], which investigated the case of the Brownian motion.
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