Asymptotics for Risk Measures of Extreme Risks

Abstract

This thesis focuses on measuring extreme risks in insurance business. We mainly use extreme value theory to develop asymptotics for risk measures. We also study the characterization of upper comonotonicity for multiple extreme risks. Firstly, we conduct asymptotics for the Haezendonck–Goovaerts (HG) risk measure of extreme risks at high confidence levels, which serves as an alternative way to statistical simulations. We split the study of this problem into two steps. In the first step, we concentrate on the HG risk measure with a power Young function, which yields certain explicitness. Then we derive asymptotics for a risk variable with a distribution function that belongs to one of the three max-domains of attraction separately. We extend our asymptotic study to the HG risk measure with a general Young function in the second step. We study this problem using different approaches and overcome a lot of technical difficulties. The risk variable is assumed to follow a distribution function that belongs to the max-domain of attraction of the generalized extreme value distribution and we show a unified proof for all three max-domains of attraction. Secondly, we study the firstand second-order asymptotics for the tail distortion risk measure of extreme risks. Similarly as in the first part, we develop the first-order asymptotics for the tail distortion risk measure of a risk variable that follows a distribution function belonging to the max-domain of attraction of the generalized extreme value distribution. In order to improve the accuracy of the first-order