个别风险模型中总索赔额分布函数的界值及模型推广研究

【作者】 刘燕；

【导师】 唐应辉；

【作者基本信息】 电子科技大学 ， 应用数学， 2003， 硕士

【摘要】 保险风险模型的研究在理论上和实际应用方面都具有十分重要的意义 ,本文对经典风险模型进行拓展、完善。并创立新的分析技术和研究方法,讨论总索赔额分布函数的界值。而后在求得总索赔额分布函数的界值的基础上,求破产概率等重要指标。1. 在经典的个别风险模型中,讨论了总索赔额分布函数的界值研究及应用,如:若单个保单索赔额分布函数F（x）有f（x）/F|-（x）关于x递增,且F（x）有均值μ（0<μ<∞）,则总索赔分布函数F_s（x）有当0≤x≤μ,F|-（x）≥e^-x/xμ) ; 当x>μ,F|-（x）≥e^-x/xμ),其中ω为方程1-ω=e^-x/xμ)的最大根（0<ω<1）。并将经典的个别风险模型推广到非同质的个别风险模型和随机利率下的个别风险模型,使用随机化的方法将非同质的个别风险模型转化为同质的个别风险模型,使用累积函数法得到随机利率下的个别风险模型中盈余过程的分布情况,而且给出了大量实例。2. 在开放的个别风险模中,研究各重要指标的求解。重点讨论了总索赔额分布函数的界值,如:单个保单的索赔X的分布函数F（x）满足:若E（X）=λ＜∞,方差为σ²,且对x>0,有1/λ（?）F|-（u）du≤F|-（x）,则有p[1-F（x）]/[1-pF（x）]≤F|-s（x）≤pe^{-λ（1-p）x}-p（1-p）[M（x）-λx],并且在此基础上求解破产概率。更多还原

【Abstract】 Research in the risk model is very important meaning in the theory and applications. From the facts of insurance , I found the new analyse ways which can be used to study the bounds of the claim amount distribution in individual risk models. In the basis of the bounds of portfolio we gained the ruin probability in this article. 1. When the claim amount distribution is some generic character, by the new analyse ways we can study the bounds of the claim amount distribution in individual risk models. In the basis of the bounds of portfolio we gained the ruin probability and others important targets. For example if one policy’s claim distribution F（x） is increasing f（x）/F|-（x） with a finite mean μ（0<μ<∞）,, then when 0≤x≤μ,F|-（x）≥e^-x/xμ) ; when x>μ,F|-（x）≥e^-x/xμ),1-ω=e^-x/xμ)where ω is 1-ω=e^-x/xμ) （0<ω<1）。Introducing the means which transform risk models with the different claim amount distribution to risk models with the same claim amount distribution. And I give some example of the transform in this chapter. And in the risk models with interest rate, discusses the important targets. 2. In open individual risk models, discusses the bounds of claim amount distribution in individual risk models by some characters of one policy’s claim distribution. For example if one policy’s claim distribution F（x） is E（X）=λ＜∞,Var（x）=σ²,and （?）x>0,1/λ（?）F|-（u）du≤F|-（x）,thenp[1-F（x）]/[1-pF（x）]≤F|-s（x）≤pe^{-λ（1-p）x}-p（1-p）[M（x）-λx],And in the basis of the bounds of portfolio we gained the ruin probability.更多还原