  In this paper we present new theoretical results on optimal estimation of
certain random quantities based on high frequency observations of a L\'evy
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 stable L\'evy process. Another contribution of our
article is the conditional mean estimation of the local time and the occupation
time measure of a linear Brownian motion. We demonstrate that the new
estimators are considerably more efficient compared to the classical
estimators. Furthermore, we discuss pre-estimation of the parameters of the
underlying models, which is required for practical implementation of the
proposed statistics.
