Modelling and Simulation of Multivariate Extremes with an Application to Insurance Data

Specialeforsvar ved Maria Voss Skotner

Titel: Modelling and Simulation of Multivariate Extremes with an Application to Insurance Data

Abstract: The modelling of extreme events has been a topic of increasing interest over the years. Extreme value theory has been applicable in various areas such as hydrology, finance and insurance. Following the work by P. Embrechts et al, S. Resnick, T. Mikosch, L. de Haan, A. Ferreira and many others, we present methods for simulating and modelling multivariate extremes.
Working with multivariate extremes, the marginal distributions and the dependence structure are handled separately. Various classes of heavy-tailed distributions are presented and their joint relationships proved, including the class of regularly varying distributions and the class of subexponential distributions. We present a framework for estimating the marginal distributions of multivariate extremes in terms of distributional convergence of properly centered and normalized maxima.
A multivariate extension of the univariate framework allows us to investigate the dependence structure of multivariate extremes in terms of the direction and amplitude of their joint distribution. Under the assumption of distributional convergence of component-wise maxima to a Frechet limit after standardization of the marginal distributions, we are able to estimate the probabilities of events which have not yet occurred.
In addition to handling multivariate extremes, we present the concept of elliptical distributions, allowing us to easily simulate multivariate extremes, on which the proposed models can be evaluated.
Lastly, the methods discussed throughout the thesis are applied to bivariate insurance data, which consists of claims reporting losses related to building and moveable property damages.

Vejleder: Thomas Mikosch
Censor:    Yuri Geogebeur, SDU