两类离散风险模型的等价性

The Equivalence of Two Disctete Risk Models

保险公司需要对发生了事故的投保客户进行赔付 假定考虑整数倍单位时刻i,i =1,2 ,… ,在时间区间 (i- 1,i]中即使发生多起事故 ,公司都在时刻i给予赔付 ,因而在时刻i可综合地视为发生了一次事故 在n个单位时间内 ,保险公司的赔付总额可以用 2种模型来进行统计 第 1种模型称之为A型 :出了事故后立即赔付 ,第i次事故的赔付额为随机变量 ξi,取值于 (0 ,∞ )且独立同分布 ,则n个单位时间内的赔付总额为 N(n)i =1 ξi,其中N(n)是n个单位时刻上出现的事故总数 第 2种模型称为B型 :每个单位时刻均赔付 ,随机变量Xi 表示i时刻的赔付额 ,取值于 [0 ,∞ )且独立同分布 ,则n个单位时间内赔付总额为 ni=1Xi 以随机过程论的观点严格地证明了

The insurance company must pay the insureds insurance calms if accidents occur. Let us consider the integer time i, i=0,1,2,3,..., we suppose that the company pays only at time i even sevaral accidents occur in time interval (i-1,i), so we may consider one accident or no accident at time i. Up to time n, the total of insurance calms may count in two models. The first model is said to be the type of A model: the company pays immediately if accident occurs at some time, the i-th calm is a random variable ξi, i=1,2,3,..., which are (0, + ∞) valued, independent each other, and have same distribution, the total of insurance calms up to time n is N(n)∑i=l ξi, where N (n) is the sum of numbers of accidents occured up to time n. The second model is said to be the type of B model: the company pays at each time i, random variable Xi is the insurance calm at time i, i=1,2,3,..., which are [0, + ∞) valued, independent each other, and have same distribution, the total of insurance calms up to time n is n∑i=l. Xi. In this paper we proved the equivalence of two models rigorously using theory of stochastic process.