On increasing (non) stationary levy processes: an easy approach using renewal processes

Official URL: http://192.168.1.20/record=b1302921 (Table of Contents)

Increasing (non)stationary Levy processes are widely used in Operations Research and Engineering. The main areas of applications of these stochastic processes are insurance mathematics, inventory control and maintenance. Special and well known instances of these processes are (non) stationary Poisson and compound Poisson processes. Since in textbooks (increasing) Levy processes are mostly regarded as special instances of continuous time martingales the main properties of Levy processes are derived by applying general results available for martingales. However, understanding the theory of martingales requires a deep insight into the theory of stochastic processes and so it might be difficult to understand the proofs of the main properties of increasing Levy processes. Therefore the main purpose of this study is to relate increasing Levy processes to simpler stochastic processes and give simpler proofs of the main properties. Fortunately there is a natural way linking increasing Levy processes to random processes occurring within renewal theory. Using this (sample path) approach and applying properties of random processes occurring within renewal theory we are able to analyze the undershoot and overshoot random process of an increasing Levy process. By a similar approach the (asymptotic) properties of the hitting time at level r can also be derived. Next to well known results we also derive new results in this thesis. In particular we extend Lorden's inequality for the renewal function and the residual life process to both the expected hitting time and the expected overshoot of an increasing Levy process at level r.