Uniform asymptotics for finite-time ruin probability in somedependent compound risk models with constant interest rate

Yang Yang Liu Wei Lin Jinguan Zhang Yulin

(1School of Economics and Management, Southeast University, Nanjing 210096, China)(2School of Mathematics and Statistics, Nanjing Audit University, Nanjing 210029, China)(3College of Mathematics and System Science, Xinjiang University, Urumqi 830046, China)(4Department of Mathematics, Southeast University, Nanjing 210096, China)

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Uniform asymptotics for finite-time ruin probability in somedependent compound risk models with constant interest rate

Yang Yang1,2Liu Wei3Lin Jinguan4Zhang Yulin1

(1School of Economics and Management, Southeast University, Nanjing 210096, China)(2School of Mathematics and Statistics, Nanjing Audit University, Nanjing 210029, China)(3College of Mathematics and System Science, Xinjiang University, Urumqi 830046, China)(4Department of Mathematics, Southeast University, Nanjing 210096, China)

Consider two dependent renewal risk models with constant interest rate. By using some methods in the risk theory, uniform asymptotics for finite-time ruin probability is derived in a non-compound risk model, where claim sizes are upper tail asymptotically independent random variables with dominatedly varying tails, claim inter-arrival times follow the widely lower orthant dependent structure, and the total amount of premiums is a nonnegative stochastic process. Based on the obtained result, using the method of analysis for the tail probability of random sums, a similar result in a more complex and reasonable compound risk model is also obtained, where individual claim sizes are specialized to be extended negatively dependent and accident inter-arrival times are still widely lower orthant dependent, and both the claim sizes and the claim number have dominatedly varying tails.

compound and non-compound risk models; finite-time ruin probability; dominatedly varying tail; uniform asymptotics; random sums; dependence structure

(1)

In the non-compound model, whereNk=1,k≥1, the finite-time ruin probability can be simplified as

(2)

This paper aims to investigate the asymptotics for the finite-time ruin probabilities in Eqs.(1) and (2) holding uniformly for alltsuch thatλ(t) is positive. Define the setΛ={t:λ(t)>0}.

1 Preliminaries

Hereafter, all the limit relationships hold forx→∞. For two positive bivariate functionsa(x,t) andb(x,t), we writea(x,t)b(x,t) (or, equivalently,b(x,t)?a(x,t)) holds uniformly for alltin a nonempty setA, if lim sup supt∈Aa(x,t)/b(x,t)≤1; we writea(x,t)～b(x,t) holds uniformly for allt∈A, ifa(x,t)b(x,t) anda(x,t)?b(x,t). For realy, the greatest integer smaller than or equal toyis denoted by [y].

(3)

(4)

and they are said to be widely orthant dependent (WOD) if they are both WUOD and WLOD.

2 Uniform Asymptotics for Finite-Time Ruin Probabilities

(5)

for someε0>0. Then for anyT∈Λ, it holds that uniformly for allt∈Λ∩[0,T],

(6)

Lemma 1 Under the conditions of Theorem 1, for allT∈Λ, it holds that uniformly fort∈Λ∩[0,T],

(7)

Proof The proof of Lemma 1 follows the line of Theorem 1.1 in Ref.[5].

asε↓0, which implies that the desired lower bound in (6) holds. Again by Lemma 1, we obtain that uniformly for allt∈Λ∩[0,T],ψ1(x,t)≤P(Dδ(t)>x)

In the following, we study the uniform asymptotics for the finite-time ruin probability in a compound renewal risk model by the investigation of the asymptotic tail behavior of random sums. Some related results can be found in Refs.[6-8].

DenotethepartialsumbySn=X1+X2+…+Xn,n≥1.

Lemma 2 Let {Xn,n=1} be END nonnegative r.v.s with common distributionF∈Dand meanμF>0, andNbe an integer-valued r.v., independent of {Xn,n=1}, with distributionG∈Dand meanμG>0. Then

(8)

Proof For any 0<ε<1 and integerm, we divide the tail probability ofSNinto three parts:

P(N=i)=:L1+L2+L3

(9)

ByF∈Dand Theorem 1 in Ref.[9], we have that

(10)

For anym

P(Si>x) =P(Si-iμF>x-iμF)≤

where the last step usesF∈DandCis a positive constant irrespective toi. By using Theorem 1 in Ref.[9] and the dominated convergence theorem, we obtain that

(11)

(12)

Thus, combining (9) to (12), we can obtain the upper bound in (8).

Now we estimate the lower bound ofP(SN>x). For any 0<ε<1 and integerm, we have that

L1+L4

(13)

ForL4, it holds that

Hence, by the strong law of the large numbers of END r.v.s[11]andF∈D, we obtain that

(14)

Therefore, (13), (10) and (14) yield the lower bound in (8).

(15)

which, byF∈DandG∈D, implies thatH∈D. So, by (1), Theorem 1 and (15), for any fixedT∈Λ, we obtain that

ψ(x,T)

(16)

(17)

holduniformlyforallt∈Λ∩[0,T]. Note that by (15), it holds that for anyy>1,

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[3]Wang K Y, Wang Y B, Gao Q W. Uniform asymptotics for the finite-time ruin probability of a dependent risk model with a constant interest rate [J].MethodolComputApplProb, 2013, 15(1): 109-124.

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[6]Yang Y, Wang Y B, Leipus R, et al. Asymptotics for tail probability of total claim amount with negatively dependent claim sizes and its applications [J].LithMathJ, 2009, 49(3): 337-352.

[7]Yang Y, Lin J G, Huang C, et al. The finite-time ruin probability in two nonstandard renewal risk models with constant interest rate and dependent subexponential claims [J].JKoreanStatistSociety, 2012, 41(2): 213-224.

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[9]Yi L, Chen Y, Su C. Approximation of the tail probability of randomly weighted sums of dependent random variables with dominated variation [J].JMathAnalAppl, 2011, 376(1): 365-372.

[10]Yang Y, Wang K Y. Precise large deviations for dependent random variables with applications to the compound renewal risk model [J].RockyMounJMath, 2013, 43(4): 1395-1414.

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帶有常數(shù)利息率的相依復(fù)合風(fēng)險(xiǎn)模型中有限時(shí)破產(chǎn)概率的一致漸近性

楊 洋1,2劉 偉3林金官4張玉林1

(1東南大學(xué)經(jīng)濟(jì)管理學(xué)院,南京 210096) (2南京審計(jì)學(xué)院數(shù)學(xué)與統(tǒng)計(jì)學(xué)院,南京 210029) (3新疆大學(xué)數(shù)學(xué)與系統(tǒng)科學(xué)學(xué)院,烏魯木齊 830046) (4東南大學(xué)數(shù)學(xué)系,南京 210096)

考慮了2個(gè)帶有常數(shù)利息率的相依更新風(fēng)險(xiǎn)模型.首先研究了非復(fù)合風(fēng)險(xiǎn)模型,其中索賠額是上尾漸近獨(dú)立且?guī)в锌刂谱儞Q尾分布的非負(fù)隨機(jī)變量,索賠時(shí)間間隔是寬下象限相依的,保費(fèi)收入過(guò)程是一個(gè)非負(fù)的隨機(jī)過(guò)程,利用風(fēng)險(xiǎn)理論中的方法,得到了有限時(shí)破產(chǎn)概率在某個(gè)有界區(qū)間上的一致漸近性.在此基礎(chǔ)上,利用隨機(jī)和尾漸近性的分析方法,進(jìn)一步研究獲得了更為復(fù)雜且合理的復(fù)合相依更新風(fēng)險(xiǎn)模型中有限時(shí)破產(chǎn)概率的一致漸近性公式,其中單個(gè)索賠額特殊化為廣義負(fù)相依的,并且事故時(shí)間間隔仍然保持寬下象限相依的,索賠額和索賠次數(shù)均為控制變換尾的.

復(fù)合及非復(fù)合風(fēng)險(xiǎn)模型;有限時(shí)破產(chǎn)概率;控制變換尾;一致漸近性;隨機(jī)和;相依結(jié)構(gòu)

O211.4

s：The National Natural Science Foundation of China (No. 11001052, 11171065, 71171046), China Postdoctoral Science Foundation (No. 2012M520964), the Natural Science Foundation of Jiangsu Province (No. BK20131339), the Qing Lan Project of Jiangsu Province.

：Yang Yang, Liu Wei, Lin Jinguan, et al. Uniform asymptotics for finite-time ruin probability in some dependent compound risk models with constant interest rate[J].Journal of Southeast University (English Edition),2014,30(1):118-121.

10.3969/j.issn.1003-7985.2014.01.022

10.3969/j.issn.1003-7985.2014.01.022

Received 2013-08-29.

Biography：Yang Yang (1979—), male, doctor, associate professor, yyangmath@gmail.com.