Past lectures

Abstract

Point processes form a powerful tool for handling spatial random events and high dimensional data sets. The course starts with an intuitive introduction to Poisson point processes with many examples and a number of applications from risk theory and financial mathematics: Lévy processes, extremal processes, and Markov jump processes with finite dimensional state space. This leads to a more formal part which treats thedistribution of a point process, Laplace functionals, and weak convergence of point processes and support functionals.

The main part of the course is based on current research of Paul Embrechts and myself, and is concerned with the problem of obtaining usefulasymptotics for the behaviour of a sample cloud at its edge. In the univariate situation this leads to a Poisson point process whose intensity is an exponential or a power function, yielding the well known Pareto distributions for exceedances, and the extreme value distributions formaxima. In the multivariate situation the intensity has a large group ofsymmetries. The algebra needed to handle such groups is developed in thecourse. For applications it is the domains of attraction which are of greatest interest. Multivariate risk theory is an active area of research, and some of the open problems in this area will be discussed.