Optimal estimation of some random quantities of a L\'evy process

TL;DR

This paper develops new theoretical estimators for key quantities of Levy processes, demonstrating their efficiency and discussing practical parameter estimation methods for high-frequency data analysis.

Contribution

It introduces novel asymptotic estimators for supremum, infimum, local time, and occupation time of Levy processes, improving efficiency over classical methods.

Findings

New estimators are more efficient than classical ones.

Asymptotic theory established for high-frequency observations.

Practical methods for parameter pre-estimation are discussed.

Abstract

In this paper we present new theoretical results on optimal estimation of certain random quantities based on high frequency observations of a L\'evy process. More specifically, we investigate the asymptotic theory for the conditional mean and conditional median estimators of the supremum/infimum of a linear Brownian motion and a stable L\'evy process. Another contribution of our article is the conditional mean estimation of the local time and the occupation time measure of a linear Brownian motion. We demonstrate that the new estimators are considerably more efficient compared to the classical estimators. Furthermore, we discuss pre-estimation of the parameters of the underlying models, which is required for practical implementation of the proposed statistics.