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

In this paper we present new theoretical results on optimal estimation of certain random quantities based on high frequency observations of a Levy 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 Levy 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.