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2022 Optimal estimation of the supremum and occupation times of a self-similar Lévy process
Jevgenijs Ivanovs, Mark Podolskij
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Electron. J. Statist. 16(1): 892-934 (2022). DOI: 10.1214/21-EJS1928

Abstract

In this paper we present new theoretical results on optimal estimation of certain random quantities based on high frequency observations of a Lévy 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 strictly stable Lévy process. Another contribution of our article is the conditional mean estimation of the local time and the occupation time of a linear Brownian motion. We demonstrate that the new estimators are considerably more efficient compared to the classical estimators studied in e.g. [6, 14, 29, 30, 38]. Furthermore, we discuss pre-estimation of the parameters of the underlying models, which is required for practical implementation of the proposed statistics.

Funding Statement

The financial support of ERC Consolidator Grant 815703 “STAMFORD: Statistical Methods for High Dimensional Diffusions” and Sapere Aude Starting Grant 8049-00021B “Distributional Robustness in Assessment of Extreme Risk” is gratefully acknowledged.

Acknowledgments

We would like to express our gratitude to Johan Segers for his comments concerning pre-estimation of model parameters.

Citation

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Jevgenijs Ivanovs. Mark Podolskij. "Optimal estimation of the supremum and occupation times of a self-similar Lévy process." Electron. J. Statist. 16 (1) 892 - 934, 2022. https://doi.org/10.1214/21-EJS1928

Information

Received: 1 July 2021; Published: 2022
First available in Project Euclid: 28 January 2022

Digital Object Identifier: 10.1214/21-EJS1928

Subjects:
Primary: 60F05 , 62G20 , 62M05
Secondary: 60G18 , 60G51 , 62G15

Keywords: Conditioning to stay positive , Lévy processes , Local time , occupation time , optimal estimation , self-similarity , supremum , weak limit theorems

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Vol.16 • No. 1 • 2022
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