Abstract

Part I
In this thesis, we first introduce and review some fluctuation theory of Levy processes, especially for general spectrally negative Levy processes and for spectrally negative Levy taxed processes. Then we consider a more realistic model by introducing draw-down time, which is the first time a process falls below a predetermined draw-down level which is a function of the running maximum. Particularly, we present the expressions for the classical two-sided exit problems for these processes with draw-down times in terms of scale functions. We also find the expressions for the discounted present values of tax payments with draw-down time in place of ruin time. Finally, we obtain the expression of the occupation times for the general spectrally negative Levy processes to spend in draw-down interval killed on either exiting a fix upper level or a draw-down lower level.



Part II
Entropy has become more and more essential in statistics and machine learning. A large number of its applications can be found in data transmission, cryptography, signal processing, network theory, bio-informatics, and so on. Therefore, the question of entropy estimation comes naturally. Generally, if we consider the entropy of a random variable knowing that it has survived up to time $t$, then it is defined as the residual entropy. In this thesis we focus on entropy and residual entropy estimation for nonnegative random variable. We first present a quick review on properties of popular existing estimators. Then we propose some candidates for entropy and residual entropy estimator along with simulation study and comparison among estimators.