Multivariate Tail Distortion Risk Measures For Elliptical Distributions

In a portfolio risk management,risks in a portfolio are often dependent.To measure an individual risk in the portfolio,such as determining the premium of the individual risk or allocating a capital to the individual risk,we should consider the dependence between the risk and other risks in the portfolio and the impacts of some extreme events on the risk measures of the risks.In this case,the risk measure of single variable is not enough,a multivariate risk measure of a portfolio of risks or a random vector is a useful tool for measuring the risks in a portfolio,multivariate tail distortion risk measure is one of then.In this thesis,we study the concept and properties of multivariate tail distortion(MTP)risk measures,we also focus on the tail distortion risk measure for elliptical distributions.The reason we choose elliptical distributions is that elliptical distributions have good prop-erties,are handled conveniencely,and are used in financial risk abroadly.In the second chapter,we introduce the definitions,properties,some kinds of elliptical distributions which are used abroadly,and introduce Choquet integrals and their properties.In the third chap-ter,we introduce the definitions,and discuss properties and conclusions that have studied of multivariate tail distortion risk measures.In the fourth chapter,combining with the prop-erties of elliptical distributions,we introduce the formulas of the tail distortion risk measure for the common tail regions (?)In particular,we consider the normal distribution important and popular in actuarial science and finance,we discuss the applications of the multivariate tail distortion risk measures in capital allocations for a portfolio of risks which have an multivariate normal distribution and explore the impact of the dependence between risks and extreme tail events in a portfolio in capital allocations.In the fifth chapter,we draw a conclusion to paper.