相依风险下的完全混合与随机变量和的最小Value-at-Risk

Complete Mixability and Minimum Value-at-Risk of Sum of Dependent Variable

摘要: 由于在金融行业中风险一般是以投资组合的形式或者在保险行业是以保单加和的形式存在，并且他们之间并不是独立的，并且相依结构大部分都是未知的，我们不能再用单一风险或者随机变量之间相互独立的方法进行研究，我们需要考虑他们之间的相依结构。在量化风险度量领域，对相依关系不确定情况下风险加和的研究很多，其中主要集中在加和的风险度量。本文主要针对在给定相同的边际分布的情况下，对VaRα(S)(Value-at-Risk)风险度量的相依结构进行了研究，从而得到了在密度单调情形下使得风险加和的VaRα(S) 最小的相依结构，这种相依结构就是在随机变量完全混合时的相依结构，最后本文得到了特殊情况，即两个随机变量完全混合的相依结构及VaRα(S) 的最大值和最小值。

Abstract: In general, the risk is in forms of portfolio in finance or in forms of the sum of insurance policy in insurance. And they are generally not independent and we do not know the correlation, and we call this correlation copula, of them. So we cannot research that using the method of investigating only one risk or regard random variables as independent. We need to consider the correlation of n random variables with given identical marginal distributions. In the quantitative risk management, the research about the sum of n random variables under uncertain correlation, especially focuses on the risk measure of the sum of n random variables. In this paper, we investigate VaRα(S) (Value-at-Risk) and the copula that it can minimize the VaRα(S) in a confident level α . Then we get the copula of minimum of VaRα(S) under monotone density. And this copula is that n random variables are complete mixable. Finally, this paper gets the minimum and maximum of VaRα(S) at the situation of two dimensions. This result is also got under complete mixability.