Lars Norvang Andersen & Soren Asmussen 
Levy Matters V [PDF ebook] 
Functionals of Levy Processes

This three-chapter volume concerns the distributions of certain functionals of Levy processes. The first chapter, by Makoto Maejima, surveys representations of the main sub-classes of infinitesimal distributions in terms of mappings of certain Levy processes via stochastic integration. The second chapter, by Lars Norvang Andersen, Soren Asmussen, Peter W. Glynn and Mats Pihlsgard, concerns Levy processes reflected at two barriers, where reflection is formulated a la Skorokhod. These processes can be used to model systems with a finite capacity, which is crucial in many real life situations, a most important quantity being the overflow or the loss occurring at the upper barrier. If a process is killed when crossing the boundary, a natural question concerns its lifetime. Deep formulas from fluctuation theory are the key to many classical results, which are reviewed in the third chapter by Frank Aurzada and Thomas Simon. The main part, however, discusses recent advances and developments in the setting where the process is given either by the partial sum of a random walk or the integral of a Levy process.