高珊1,2

(1.中南大學(xué)數(shù)學(xué)科學(xué)與計(jì)算技術(shù)學(xué)院,湖南長沙 410075; 2.阜陽師范學(xué)院數(shù)學(xué)與計(jì)算科學(xué)學(xué)院,安徽阜陽 236041)

具有紅利邊界的Erlang(2)風(fēng)險(xiǎn)模型

高珊1,2

(1.中南大學(xué)數(shù)學(xué)科學(xué)與計(jì)算技術(shù)學(xué)院,湖南長沙 410075; 2.阜陽師范學(xué)院數(shù)學(xué)與計(jì)算科學(xué)學(xué)院,安徽阜陽 236041)

給出了具有邊界紅利策略的Erlang(2)風(fēng)險(xiǎn)模型,在此紅利策略下，若保險(xiǎn)公司的盈余在紅利線以下時(shí)不支付紅利,否則紅利以低于保費(fèi)率的常速率予以支付.對(duì)于該模型，本文推導(dǎo)了Gerber-Shiu折現(xiàn)懲罰函數(shù)所滿足的兩個(gè)積分-微分方程和更新方程．

Erlang(2)風(fēng)險(xiǎn)過程；折現(xiàn)懲罰函數(shù)；積分-微分方程；紅利策略

1 引言

邊界策略最初是De Finetti(1957)對(duì)二項(xiàng)模型提出的,最近關(guān)于復(fù)合Poisson風(fēng)險(xiǎn)模型更一般的邊界策略得到廣泛的研究[1-7].本文考慮具有如下邊界紅利策略的Erlang(2)風(fēng)險(xiǎn)模型:

δ≥0可以代表利息強(qiáng)度,I(…)表示示性函數(shù),w(x1,x2)是關(guān)于x1≥0和x2＞0的非負(fù)有界函數(shù).

2 關(guān)于mb(u)的積分-微分方程

在這部分我們將推導(dǎo)關(guān)于mb(u)的兩個(gè)積分-微分方程,一個(gè)是當(dāng)初始盈余低于邊界的時(shí)候,另一個(gè)是初始盈余高于邊界.令

3 關(guān)于m2(u)的更新方程

[1]A lbrecher H,Hartinger J,Tichy R F.On the distribution of dividend paym ents and the discounted penalty function in risk modelwith linear dividend barrier[J].Scandinavian Actuarial Journal,2005,2:103-126.

[2]Gerber H U,Shiu E SW.On op tim al dividend strategies in the com pound Poissonmodel[J].North Am erican Actuarial Journal,2006,10(2):76-93.

[3]Lin X S,W illm ot G E,D rekic S.The classical risk model with a constant dividend barrier:analysis of the Gerber-Shiu discounted penalty function[J].Insurance:Mathem atics and Econom ics,2003,33(3):551-566.

[4]Gerber H U,Shiu E SW.The time value of ruin in a Sparre Andersen model[J].North American Actuarial Journal,2005b,92:49-84.

[5]Lin X S,Pavlova K P.The com pound Poisson risk model with a threshold dividend strategy[J].Insurance:Mathem atics and Econom ics,2006,38:57-80.

[6]Li S,Garrido J.On a class of renewal riskmodelswith a constant dividend barrier[J].Insurance:Mathematics and Econom ics,2004,35:691-701.

[7]方世祖,羅建華.雙復(fù)合Poisson風(fēng)險(xiǎn)模型[J].純粹數(shù)學(xué)與應(yīng)用數(shù)學(xué),2006,22(2):271-278.

[8]Li S,Garrido J.On ruin for Erlang(n)risk process[J].Insurance:Mathem atics and Econom ics,2004,34:391-408.

[9]D ickson D C M,Hipp C.On the time of ruin for Erlang(2)risk process[J].Insurance:Mathem atics and Econom ics,2001,29:333-344.

[10]Lin X S.Discussion of Y.Cheng and Q.Tang’s Moments of the surp lus before ruin and the deficit at ruin inthe Erlang(2)risk p rocess[J].North Am erican Actuarial Journal,2003,7(3):122-124.

The Erlang(2)risk model with adividend barrier

GAO Shan1,2

(1.School of Mathem atics,Central South University,Changsha 410075,China; 2.Department of Mathematics,Fuyang Normal College,Fuyang 236041,China)

In this paper,we present the Erlang(2)risk model with a dividend barrier strategy.Under such strategy,no dividends are paid if the insurer’s surp lus is below certain barrier level,when the surp lus is above this barrier level,dividends are paid at a constant rate that does not exceed the prem ium rate.For the risk model,two integro-differentialequations and a renewalequation for the Gerber-Shiu discounted penalty function are derived.

Erlang(2)risk model,discounted penalty function,integro-differential equation,dividend strategy 2000M SC:60K 20,91B30

O211.67

A

1008-5513(2009)02-0251-07

2007-09-04.

安徽省高等學(xué)校省級(jí)自然科學(xué)研究項(xiàng)目(KJ2007B 183).

高珊(1975-),博士,講師,研究方向:隨機(jī)過程,排隊(duì)論.