Optimal estimation of the supremum and occupation times of a self-similar Lévy process∗

Conditioning to stay positive; local time; Lévy processes; occupation time; optimal estimation; self-similarity; supremum; weak limit theorems; Statistics and Probability; Statistics, Probability and Uncertainty

Abstract :

[en] 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 underly-ing models, which is required for practical implementation of the proposed statistics.

Disciplines :

Mathematics

Author, co-author :

Ivanovs, Jevgenijs; Department of Mathematics, Aarhus University, Denmark

PODOLSKIJ, Mark ; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH)

External co-authors :

yes

Language :

English

Title :

Optimal estimation of the supremum and occupation times of a self-similar Lévy process∗

Publication date :

2022

Journal title :

Electronic Journal of Statistics

eISSN :

1935-7524

Publisher :

Institute of Mathematical Statistics

Volume :

Issue :

Pages :

892 - 934

Peer reviewed :

Peer Reviewed verified by ORBi

European Projects :

H2020 - 815703 - STAMFORD - Statistical Methods For High Dimensional Diffusions

Name of the research project :

Statistical Methods For High Dimensional Diffusions

Funders :

ERC - European Research Council
Union Européenne

Funding number :

815703

Funding text :

∗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.

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