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

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