Analysis Of Risk Metrics, Random Order And Homomonotony Based On Two Types Of Distribution

Risk is around of ius. In our daily life, there are a lot, of uncertain events happening,bring losses to us. In order to measure these risks, we often use VaR, TVaR or other risk measures to measure risks. However, these risk measures depend on distribution. The elliptical distribution and log- elliptical distribution are often used in our daily life. So study based on these two distributions are presented in this paper. Stochastic order is an important.tool in probability and statistics, especially in risk decisions.In this thesis, the first part of the second chapter introduces the definition and property of elliptical distribution and log-elliptical distribution. The second part introduces the defi-nition of comonotonicity, and when random vector is multivariate elliptical distributions or multivariate log-elliptical distributions, if it is comonotonic, what the conditions should be satisfy. The third chapter of this thesis introduces the definition and property of risk and risk measure. We also give the form of some risk measures when risk is elliptical distribution or log-elliptical distribution. The forth section introduces the definition of stochastic order,convex order, liner convex order and so on. We then discuss that if two random variables are elliptical distributions and log-elliptical distributions, the relationships between the mean and the variance if they satisfy some stochastic orders. Moreover, extensions to multivariate distributions are discussed.