郭曉莉,文麗壹
(重慶理工大學(xué) 數(shù)學(xué)與統(tǒng)計(jì)學(xué)院,重慶 400054)
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保險(xiǎn)和金融風(fēng)險(xiǎn)相依的破產(chǎn)概率研究
郭曉莉,文麗壹
(重慶理工大學(xué) 數(shù)學(xué)與統(tǒng)計(jì)學(xué)院,重慶400054)
摘要:考慮一家保險(xiǎn)公司暴露于保險(xiǎn)風(fēng)險(xiǎn)和金融風(fēng)險(xiǎn)兩種風(fēng)險(xiǎn)環(huán)境,分別用兩組隨機(jī)變量量化這兩種風(fēng)險(xiǎn),用離散時(shí)間風(fēng)險(xiǎn)模型表述保險(xiǎn)公司盈余過(guò)程,研究了在保險(xiǎn)風(fēng)險(xiǎn)和金融風(fēng)險(xiǎn)漸近獨(dú)立相依假設(shè)下的保險(xiǎn)公司有限時(shí)間破產(chǎn)概率問(wèn)題。當(dāng)保險(xiǎn)風(fēng)險(xiǎn)的分布屬于次指數(shù)分布族或長(zhǎng)尾分布族時(shí),分別推導(dǎo)了有限時(shí)間破產(chǎn)概率的漸近等價(jià)關(guān)系式,這將簡(jiǎn)化保險(xiǎn)公司在風(fēng)險(xiǎn)評(píng)估中的計(jì)算問(wèn)題。
關(guān)鍵詞:保險(xiǎn)風(fēng)險(xiǎn);金融風(fēng)險(xiǎn);破產(chǎn)概率;次指數(shù);長(zhǎng)尾
考慮一個(gè)離散時(shí)間保險(xiǎn)風(fēng)險(xiǎn)模型。設(shè)保險(xiǎn)公司的初始資本金為x≥0,保險(xiǎn)公司第k年內(nèi)的凈收入由實(shí)值隨機(jī)變量Ak(k=1,2,…,n)表示,保險(xiǎn)公司在第k年將資金投入到有風(fēng)險(xiǎn)和無(wú)風(fēng)險(xiǎn)市場(chǎng)所產(chǎn)生的綜合收益率由非負(fù)隨機(jī)變量Bk(k=1,2,…,n)表示。假設(shè)保險(xiǎn)公司的凈收入Ak在第k年的年末計(jì)算,則保險(xiǎn)公司在第k年末的盈余值Uk滿足以下遞歸等式:
(1)
由式(1)遞歸可以得到:
(2)
(3)
(4)
保險(xiǎn)公司有限時(shí)間內(nèi)的破產(chǎn)概率ψ(x;n)可重新表示為:
(5)
在實(shí)際應(yīng)用中,直接計(jì)算保險(xiǎn)公司在有限時(shí)間內(nèi)的破產(chǎn)概率ψ(x;n)是相當(dāng)繁瑣甚至不現(xiàn)實(shí)的,因此本文主要研究當(dāng)x→∞時(shí)ψ(x;n)的漸近等價(jià)表達(dá)式。在現(xiàn)有文獻(xiàn)研究中,{Xk;k=1,2,…,n}和{Yk;k=1,2,…,n}分別被稱為保險(xiǎn)風(fēng)險(xiǎn)和金融風(fēng)險(xiǎn)。離散時(shí)間保險(xiǎn)風(fēng)險(xiǎn)模型的破產(chǎn)概率問(wèn)題已經(jīng)被廣泛研究。根據(jù)保險(xiǎn)風(fēng)險(xiǎn)和金融風(fēng)險(xiǎn)相依關(guān)系分類,研究方向主要分以下3類:① {Xk;k=1,2,…,n}和{Yk;k=1,2,…,n}都是獨(dú)立隨機(jī)變量序列,且兩序列之間獨(dú)立,如文獻(xiàn)[1-4];② {Xk;k=1,2,…,n}和{Yk;k=1,2,…,n}中至少有一組是相依的,但兩序列之間獨(dú)立,見(jiàn)文獻(xiàn)[5-7];③ {Xk;k=1,2,…,n}和{Yk;k=1,2,…,n}兩序列之間存在相依關(guān)系,見(jiàn)文獻(xiàn)[8-11]。
1預(yù)備知識(shí)和主要結(jié)論
本文除非另有說(shuō)明,所有的極限關(guān)系都是在x→∞時(shí)成立。對(duì)于兩個(gè)正函數(shù)f(·)和g(·),若limf(x)/g(x)=1,則記為 f(·)~g(·);若liminff(x)/g(x)≥1,則記為 f(x)g(x);若limsupf(x)/g(x)≤1,則記為f(x)g(x)。除此之外,定義x+=max{x,0}為實(shí)數(shù)x的正部。
1.1重尾分布族
這一部分給出重尾及兩類重尾分布族的定義,其余重尾分布族的介紹可參看文獻(xiàn)[12]。
定義1稱隨機(jī)變量X或者它的分布函數(shù)V是重尾的,如果對(duì)任意的s>0滿足
這里把重尾分布族記為K。
相反,稱隨機(jī)變量X或者它的分布函數(shù)V是輕尾的,如果存在一個(gè)s0>0,使得對(duì)所有0
定義2稱定義在R上的分布函數(shù)V屬于長(zhǎng)尾分布族,若對(duì)任意的y≠0,滿足
記作V∈L。
定義3稱定義在R+上的分布函數(shù)V屬于次指數(shù)分布族,若對(duì)某任意的n≥2,滿足
記作V∈S。
在一般情況下,若定義在R上的分布函數(shù)V滿足V+(x)=V(x)I(x≥0)∈S,則也稱V∈S。
假設(shè)隨機(jī)變量X1,…,Xn獨(dú)立同分布,且分布函數(shù)V∈S,則有
顯然S?L,因?yàn)閷?duì)任意定義在R+上的分布函數(shù)V∈S,x≥y>0,有
則當(dāng)x→∞,V(x)-V(y)≠0時(shí),
故V∈L。
1.2相依結(jié)構(gòu)
考慮本文的離散時(shí)間保險(xiǎn)風(fēng)險(xiǎn)模型,對(duì)k=1,2,…,n,實(shí)值隨機(jī)變量Xk滿足下面的相依結(jié)構(gòu):
顯然,假設(shè)A包含以下兩種情形:
常見(jiàn)的FGM相依也滿足漸近獨(dú)立條件。因?yàn)槿綦S機(jī)變量X1,…,Xn服從n維FGM分布:
1.3主要結(jié)論
(6)
(7)
2引理及證明
首先給出3個(gè)基本的引理,這3個(gè)引理在結(jié)果的證明中將會(huì)被用到。其中:引理1來(lái)自Cline[14]的推論2.5;引理2是文獻(xiàn)[2]中引理3.2的一個(gè)特例,即r=0的情形;引理3源自文獻(xiàn)[2]中引理3.10。針對(duì)所有引理,這里不再給出證明。
引理1如果隨機(jī)變量X的分布函數(shù)F∈S,隨機(jī)變量Y有界,則XY對(duì)應(yīng)的分布函數(shù)H∈S。
因?yàn)镕k∈S,θk有界,則由引理1得Hk∈S。
H1*H2∈S且Pr(Z1+Z2>x)~Pr(Z1>x)+Pr(Z2>x)
H1*H2*H3∈S且Pr(Z1+Z2+Z3>x)~Pr(Z1>x)+Pr(Z2>x)+Pr(Z3>x)
以此類推,得H1*…*Hn∈S且Pr(Z1+…+Zn>x)~Pr(Z1>x)+…+Pr(Zn>x),即
Pr(Zi>xi,Zj>xj)=o(1)Pr(Zj>xj)
因此:
又因?yàn)?/p>
故定理1得證。
根據(jù)文獻(xiàn)[6]引理4.1,由數(shù)學(xué)歸納法可得Sn的分布函數(shù)屬于L,即對(duì)某個(gè)A≠0,有
其中,M′是θl的上界。故定理2得證。
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(責(zé)任編輯劉舸)
The Ruin Probability with Dependent Insurance and Financial Risks
GUO Xiao-li, WEN Li-yi
(Department of mathematics and statistics, Chongqing University of Technology,Chongqing 400054, China)
Abstract:Considering an insurance company exposed to an environment that contains insurance and financial risks, these two kinds of risks were quantified by two sets of random variables and the surplus process of the insurance company was described by a discrete-time risk model. This paper investigated the finite-time ruin probability with asymptotic independence insurance and financial risks. When the distributions of the insurance risk belong to the subexponential distribution class or the long-tailed distribution class, we derived some asymptotic equivalent relationships for the finite-time ruin probability, respectively. These will simplify the calculation of the insurance companies in the risk assessment.
Key words:insurance risk; financial risk; ruin probability; subexponentiality; long tail
收稿日期:2015-12-16
基金項(xiàng)目:國(guó)家社會(huì)科學(xué)基金資助項(xiàng)目(14BJY200)
作者簡(jiǎn)介:郭曉莉(1991—),女,河南周口人,碩士研究生,主要從事應(yīng)用數(shù)學(xué)研究。
doi:10.3969/j.issn.1674-8425(z).2016.05.024
中圖分類號(hào):O211.9
文獻(xiàn)標(biāo)識(shí)碼:A
文章編號(hào):1674-8425(2016)05-0135-06
引用格式:郭曉莉,文麗壹.保險(xiǎn)和金融風(fēng)險(xiǎn)相依的破產(chǎn)概率研究[J].重慶理工大學(xué)學(xué)報(bào)(自然科學(xué)),2016(5):135-140.
Citation format:GUO Xiao-li1, WEN Li-yi.The Ruin Probability with Dependent Insurance and Financial Risks[J].Journal of Chongqing University of Technology(Natural Science),2016(5):135-140.