Sparre Andersen模型中破產概率的邊界

宋會杰,李祥發,郭鵬江

(西北大學數學系,陜西西安 710069)

Sparre Andersen模型中破產概率的邊界

宋會杰,李祥發,郭鵬江

(西北大學數學系,陜西西安 710069)

研究在Andersen Sparre模型中,當破產概率的初始邊界已知的時候,根據更新方程和更新方程中函數的單調性來改進破產概率的邊界,并進一步改進了嚴重損失函數G(x,y)的邊界.

更新模型;破產概率;損失函數;更新方程

1 引言

一般情況下,φ和F是非解析的,但是當K或P是相位型分布時能寫出其顯示形式.本文首先得到ψ(x)的初始上下邊界,然后把(1.2)的積分區域分成小區間,在每個小區間上用初始上下邊界代替ψ(x?t),這樣我們就可以得到改進了的ψ(x)的上下邊界,這種方法的優點是這一過程不僅僅得到ψ(x)數值的邊界而且能用遞歸的方法得到ψ(x)解析的邊界.

2 破產概率的邊界

3 損失函數G(x,y)的邊界

4 討論

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[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]卡爾斯M,胡發茲.現代精算風險理論[M].北京:科學出版社,2005.

[4]唐啟鶴.重尾索賠下關于破產概率的一個等價式[J].中國科學,2002,32(3):261-266.

[5]孔繁超,曹龍,王金亮等.對于大索賠的平衡更新模型的破產概率[J].數學年刊:A輯,2002,23(4):531-536.

[6]成世杰.破產論研究綜述[J].數學進展,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.

陜西省科技發展計劃基金資助項目(20000K 08-G 15).

宋會杰(1981-),在讀碩士,研究方向：概率論與數理統計.

2000M SC:60G 26