宋會(huì)杰,李祥發(fā),郭鵬江

(西北大學(xué)數(shù)學(xué)系,陜西西安 710069)

Sparre Andersen模型中破產(chǎn)概率的邊界

宋會(huì)杰,李祥發(fā),郭鵬江

(西北大學(xué)數(shù)學(xué)系,陜西西安 710069)

研究在Andersen Sparre模型中,當(dāng)破產(chǎn)概率的初始邊界已知的時(shí)候,根據(jù)更新方程和更新方程中函數(shù)的單調(diào)性來改進(jìn)破產(chǎn)概率的邊界,并進(jìn)一步改進(jìn)了嚴(yán)重?fù)p失函數(shù)G(x,y)的邊界.

更新模型;破產(chǎn)概率;損失函數(shù);更新方程

1 引言

一般情況下,φ和F是非解析的,但是當(dāng)K或P是相位型分布時(shí)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(qū)域分成小區(qū)間,在每個(gè)小區(qū)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(jìn)了的ψ(x)的上下邊界,這種方法的優(yōu)點(diǎn)是這一過程不僅僅得到ψ(x)數(shù)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(chǎn)概率的邊界

3 損失函數(shù)G(x,y)的邊界

4 討論

[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]卡爾斯M,胡發(fā)茲.現(xiàn)代精算風(fēng)險(xiǎn)理論[M].北京:科學(xué)出版社,2005.

[4]唐啟鶴.重尾索賠下關(guān)于破產(chǎn)概率的一個(gè)等價(jià)式[J].中國科學(xué),2002,32(3):261-266.

[5]孔繁超,曹龍,王金亮等.對(duì)于大索賠的平衡更新模型的破產(chǎn)概率[J].數(shù)學(xué)年刊:A輯,2002,23(4):531-536.

[6]成世杰.破產(chǎn)論研究綜述[J].數(shù)學(xué)進(jìn)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(fā)展計(jì)劃基金資助項(xiàng)目(20000K 08-G 15).

宋會(huì)杰(1981-),在讀碩士,研究方向：概率論與數(shù)理統(tǒng)計(jì).

2000M SC:60G 26