Two Issues Related To Dependency Risk

In recent years, the issue of dependent risks has received remarkable attention in the risk theory ?eld after the theoretical system of independent risks has been basically perfected.There is no doubt that we have to face the issue of dependent risks. Therefore, it’s of signi?cance to deal with the methods of measuring the dependent risks, to know how the dependence changes the ruin probability and so on. In this paper, we consider two problems about dependent risks. One is about a new dependent measure, and another one is on ordered ruin probabilities of multidimensional risk models when various types of claims are dependent and claim arrival processes are dependent at the same time. The tools we used contain copula function, comonotonicity theory, supermodular order and so on. According to the contents, the thesis can be divided into two chapters:(1) A new multivariate dependence measure based on comonotoncity.In this chapter, we introduce a new multivariate dependence measure based on comonotonicity by means of product moment which motivated by the recent papers of Koch and Schepper(ASTIN Bulletin 2011, 41: 191-213) and Dhaene et al.(Journal of Computational and Applied Mathematics 2014, 263: 78-87). The new dependence measure has many good properties including the axioms of normalization, monotonicity, permutation invariance and duality. Some difference and relation between the new dependence measure and other multivariate measures are analyzed by some examples. When m = 2 our new dependence measure is similar to the classical ones while m ≥ 3 it differs from them. In addition, we give the estimation of the new measure.(2) On ordered ruin probabilities of multidimensional risk models.In this chapter, we consider the orders of the ruin probabilities of multidimensional risk models based on the dependence among various types of claims and claim arrival processes.It generalizes the model in Cai and Li(Journal of Multivariate Analysis, 2007, 98(4):757-773). We focus on three common ruin probabilities in multidimensional compound risk models and using the comparison methods to show how some ruin probabilities increase,whereas the others decrease, as the dependence of claims and arrival processes grow. Consequently, several simple bounds are obtained based on the comonotonicity theory. The proposed method may be useful for the model perturbed by a multidimensional Brownian motion, which provides us an idea to do future work.