Herbstsemester 2012
Andreas Kyprianou
Gerber-Shiu risk theory through Lévy processes


I will give a complete review of classical and modern insurance risk theory through the eyes of excursion theory for Lévy processes. To keep the technical requirements to a minimum, the course will deal largely with the case of the classical Cramér-Lundberg process, developing in detail the Poissonian structure of sojourns from the minimum, moving towards the end of the course into a more general Lévy set-up. The objective is to go far beyond the classical ruin problems, into the realms of dividend strategies which correspond to refracted, reflected and super- and sub-reflected Lévy processes as well as focusing on the importance of the modern theory of scale functions for spectrally negative Lévy processes in the analysis. Much of what will be presented will cover, at the appropriate level, the main developments that have occurred in the last 5-10 years in the research literature. If there is time, I would also like to deal with some completely new Monte-Carlo simulation methods. The course will assume core basic knowledge of Markov processes, knowledge of measure theoretic probability as well as core facts from analysis. The course will break roughly into the following sections.

Part 1: Basic tools and analysis.
- Poisson fields and Poisson processes.
- The Cramér-Lundberg process and variants thereof.
- The Laplace exponent and its Bernstein factor.
- Poisson process of excursions from the minimum.
- Scale functions, renewal theory and ladder heights.
- Tractable examples.
- The ruin problem reformulated.

Part 2: Gerber-Shiu theory.
- The Gerber-Shiu problem.
- Dividend problems.
- Reflection strategies.
- Refraction strategies.
- Perturbed processes and tax.

Part 3: residual material (if there is time).
- Working with more general Lévy processes.
- The importance and difficulties of the Wiener-Hopf factorisation.
- Simulating the paths of Lévy processes using the Wiener-Hopf factorisation.

Time:             Mi 10-12
Auditorium:  HG G 43 (HWZ)
Begins:          26. September

M. Struwe