宋会杰,李祥发,郭鹏江

(西北大学数学系,陕西西安 710069)

Sparre Andersen模型中破产概率的边界

宋会杰,李祥发,郭鹏江

(西北大学数学系,陕西西安 710069)

研究在Andersen Sparre模型中,当破产概率的初始边界已知的时候,根据更新方程和更新方程中函数的单调性来改进破产概率的边界,并进一步改进了严重损失函数G(x,y)的边界.

更新模型;破产概率;损失函数;更新方程

1 引言

一般情况下,φ和F是非解析的,但是当K或P是相位型分布时能写出其显示形式.本文首先得到ψ(x)的初始上下边界,然后把(1.2)的积分区域分成小区间,在每个小区间上用初始上下边界代替ψ(x−t),这样我们就可以得到改进了的ψ(x)的上下边界,这种方法的优点是这一过程不仅仅得到ψ(x)数值的边界而且能用递归的方法得到ψ(x)解析的边界.

2 破产概率的边界

3 损失函数G(x,y)的边界

4 讨论

[1]Politis K S.Bounds for the probability and severity of ruin in the Sparre Andersen model[J].Mathem atics and Econom ics,2005,36:165-177.

[2]Chajiconstantinidis ST,Politis K S.Two-sidesbounds for the distribution of the deficit at ruin in the renewal risk model[J].Mathem atics and Econom ics,2007,41:41-51.

[3]卡尔斯M,胡发兹.现代精算风险理论[M].北京:科学出版社,2005.

[4]唐启鹤.重尾索赔下关于破产概率的一个等价式[J].中国科学,2002,32(3):261-266.

[5]孔繁超,曹龙,王金亮等.对于大索赔的平衡更新模型的破产概率[J].数学年刊:A辑,2002,23(4):531-536.

[6]成世杰.破产论研究综述[J].数学进展,2002,31(5):404-422.

Bounds for the probab ility and severity of ruin in the sparre Andersen model

SONG Hui-jie,LIXiang-fa,GUO Peng-jiang

(Departm ent of Mathem atics,Northwest University,Xi’an 710069,China)

We propose a method for im proving existing upper and lower bounds for ruin probabilities in the Sparre Andersen model.Them ethod is based on the renewal equations for the probability of ruin in that model and m ay be used recursively.We discuss how the m ethod can be used m ore generally for solutions of renewal equations,provided that the function in question ismonotone and initial bounds for it are availab le.We illustrate this by considering the severity of ruin in the Sparre Andersen model,for which we obtain lower and upper bounds;the app lication of them ethod is then straightforward.

sparre Andersen model,ruin probability,severity of ruin,renewal equation

O212

A

1008-5513(2009)02-0408-06

2008-06-27.

陕西省科技发展计划基金资助项目(20000K 08-G 15).

宋会杰(1981-),在读硕士,研究方向：概率论与数理统计.

2000M SC:60G 26