Studies On Comonotonic Risks And Related Problems

In insurance, when more than one risks are involved they are usually assumed to be independent. For example, in collective risk model, Panjer's recursive formula and De Peril's recursive formula which are used to determine the distribution of aggregate claim are based on the independent assumption. Another example is the multiple-life model in which the future lifetimes of the insureds are assumed to be independent in classic textbooks. But due to some common influence, risks are usually related to each other more or less, so it is valuable to study the dependence among risks, and studies on the risk model under positively dependent situation is a focus in actuarial science currently. Chapter 2 of this paper describes a special dependent structure-comonotonic structure. In this chapter some results about the equivalence of a series of definitions of comonotonicity and the distribution of the sum of comonotonic random variales are from Dhaene et al(2002a); We point out that the statement that all consecutive components in a random vector are comonotonic implies the the comonotonicity of the vector if all components except the first and the last are continous r.v.'s, and that comonotonicity is invariate under non-decreasing transforms; We also study the comonotonicity of mixed random variables. In chapter 3 we introduce correlation order and two types of positive dependence: Positive Quadrant Dependence, Positive Regression Dependence. We also study how they are related to comonotonicity and three correlation coefficients. In chapter 4, assuming that distributions of all individual risks are fixed, we introduce the upper and lower bounds of the aggregate claims in the sense of convex order; We also point out that as long as common hazard exists, the independence assumption will underestimate the danger of the aggregate claim. Chapter 5 is an application of comonotonicy in two-life model. When assuming the future lifetimes of the two insureds are comonotonic we obtain the distributions of the future lifetimes of the joint-life status and the last-to-survivor status, which is similar to those when assuming independce. When the future lifetimes of the two insureds are PQD, we give a easy and feasible method to construct life tables of the joint-life status and the last-to-survivor status.The topic of chapter 6 is the fully discrete ruin model under stochastic interest rate. We obtain recursive formulas of a series of distributions(including: the distribution of ruin time T, the joint distribution of the deficit whe ruin Ut and T, the joint distribution of the surplus immediately before ruin Ut- and T, the joint distribution of Ut, Ut- and T, and the joint distribution of the time in red T and T). Based on these formulas we give some integral equations(including: the ruin probability (u), the cdf of Ut, the cdf of Ut- and the joint cdf of Ut and Ut-) and the expression of P{T = n}, n = 0,1, ....