Andreas E . Kyprianou Gerber – Shiu Risk Theory Draft

Springer Preface These notes were developed whilst giving a graduate lecture course (Nachdiplomvor-lesung) in the fall of 2012 at the Forschungsinstitut für Mathematik (FIM), ETH Zürich, Switzerland. The same course was given in the spring of 2013 at Centro de Investigación en Matemáticas (CIMAT), Guanajuato, Mexico and, simultaneously by video-link, at the Instituto de Mathemáticas, UNAM in Mexico City. The title of these lecture notes may come as surprise to some readers as, to date, the term Gerber–Shiu Risk Theory is not widely used. One might be more tempted to simply use the title Ruin theory for Cramér-Lundberg models instead. However, my objective here is to focus on the recent interaction between a large body of research literature, spear-headed by Hans Gerber and Elias Shiu, concerning ever more sophisticated questions around the event of ruin for the classical Cramér-Lundberg surplus process, and the parallel evolution of the fluctuation theory of Lévy processes. The fusion of these two fields has provided economies to proofs of older results, as well as pushing much further and faster classical theory into what one might describe as exotic ruin theory. The latter may be considered as the study of ruinous scenarios which involve perturbations to the surplus coming from dividend or tax payments that have a historical path dependence. These notes keep to the Cramér–Lundberg setting. However, the text has been written in a form that appeals to straightforward and accessible proofs, which take advantage, as much as possible , of the fact that Cramér–Lundberg processes have stationary and independent increments and no upward jumps. I would like to thank Paul Embrechts for the invitation to spend six months at the FIM and the opportunity to develop and present this material. Similarly I would like to thank Maria-Emilia Caballero, Juan Carlos Pardo and Victor Rivero for the invitation to give the same course in CIMAT and UNAM simultaneously. I would also like to thank all attendees in Zürich, Guanajuato and Mexico City for their comments (especially Whilst I was in Switzerland I had the opportunity to meet and discuss some of the material in this book with Hans Gerber and Hansjörg Albrecher for which I would also like to express my gratitude. Springer produced three anonymous referee's reports whose useful comments I am also grateful for. Finally I would like to thank v vi Preface Nick Bingham and Erik Baurdoux who have diligently attended to …