On the derivation of the distribution of the overshoot and undershoot stochastic process in increasing Lévy Processes: a renewal theory approach

Lévy processes with increasing sample paths or subordinators are widely used in Operations Research and Engineering. The main areas of applications of these stochastic processes are insurance mathematics, inventory control, maintenance and reliability theory. Special and well-known instances of these increasing processes are stationary Poisson and compound Poisson processes. Since increasing Lévy processes are mostly regarded as special instances of continuous time martingales the main properties of Lévy 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 Lévy processes. Therefore, the main purpose of this study is to relate increasing Lévy processes to simpler stochastic processes and give simpler proofs of the main properties. Fortunately, there is a natural way linking increasing Lévy 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 Lévy process. Next to well-known results we also derive new results in this paper. In particular, we extend Lorden’s inequality for the renewal function and the residual life process to the expected overshoot of an increasing Lévy process at level r.