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