GAO Miaomiao，WANG Kaiyong，CHEN Lamei，QIAN Haojun

（School of Mathematics and Physics，SUST，Suzhou 215009，China）

Asymptotics of the finite-time ruin probability of a delayed risk model perturbed by diffusion with a constant interest rate

GAO Miaomiao，WANG Kaiyong*，CHEN Lamei，QIAN Haojun

（School of Mathematics and Physics，SUST，Suzhou 215009，China）

This paper investigated a delayed and dependent risk model perturbed by diffusion with a constant interest rate,in which the claim sizes and inter-arrival times have some dependent structures,and the claim arrival times constitute a delayed quasi renewal counting process.When the distribution of the claim sizes is heavy-tailed,the asymptotics of the finite-time ruin probability of the above risk model has been obtained.

dependent risk model；constant interest rate；finite-time ruin probability；asymptotics

1 Introduction

In this paper，we consider a delayed and dependent risk model perturbed by diffusion with a constant interest rate.In this model，the claim sizes，Xn，n≥1，form a sequence of identically distributed random variables（r.v.s）with common distribution F and the inter-arrival times，Yn，n≥1，form a sequence of r.v.s.The times of successive claims，，constitute a quasi renewal counting process

where IAis the indicator function of the set A.Denote the renewal function by λ（t）=EN（t），t≥0 and assume that λ（t）＜∞ for all t≥0.The total amount of premium accumulated up to time t≥0，denoted by C（t）with C（0）=0 and C（t）＜∞ almost surely（a.s.）for every t＞0，is a nonnegative and nondecreasing stochastic process.As a perturbed term，B（t），t≥0，is a standard Brownian motion with volatility parameter σ≥0.Assume that{Xn，n≥1}，{Yn，n≥1}，{C（t），t≥0}and{B（t），t≥0}are mutually independent.Let δ≥0 be the constant interest rate，that is to say，after time t a capital y becomes yeδt.Let x≥0 be the initial capital reserve of the insurance company. Then the total reserve up to time t≥0，denoted by U（x，t），satisfieswhereis the total amount of claims up to time t≥0 with S（t）=0 when N（t）=0.Hence，the ruin probability within a finite time t＞0 is defined by

When the claim sizes，Xn，n≥1，are independent and identically distributed（i.i.d.），the inter-arrival times，Yn，n≥1，are also i.i.d.and σ=0，the ruin probabilities of this risk model have been investigated by many researchers[1-8].Recently，more attention has been paid to some dependent risk model[9-11].

Wang et al[12]introduced a new kind of dependence structure and investigated the ruin probability of a dependent risk model with this dependence structure.

Definition 1For the r.v.s{ξn，n≥1}，if there exists a finite real sequence{gU（n），n≥1}，satisfying for each n≥1 and for all xi∈（-∞，∞），1≤i≤n

then we say that the r.v.s{ξn，n≥1}are widely upper orthant dependent（WUOD）；if there exists a finite real sequence{gL（n），n≥1}satisfying for each n＞1 and for all xi∈（-∞，∞），1≤i≤n

then we say that the r.v.s{ξn，n≥1}are widely lower orthant dependent（WLOD）.

Recall that when gL（n）=gU（n）≡1 for any n≥1 in（2）and（3），the r.v.s{ξn，n≥1}are called negatively upper orthant dependent（NUOD）and negatively lower orthant dependent（NLOD），respectively，and say that the r.v.s{ξn，n≥1}are negatively orthant dependent（NOD）if{ξn，n≥1}are both NUOD and NLOD[13-14].Say that the r.v.s{ξn，n≥1}are pairwise negatively quadrant dependent（NQD）or pairwise NOD，if for all i≠j，r.v.s ξiand ξjare NOD[15].So the WUOD and WLOD structures contain some common negatively dependent r.v.s.Wang et al[12]notes that these structures also contain some positively dependent r.v.s and some other r.v.s.Proposition 1.1 of [12]presents a property about the WUOD r.v.s.

Proposition1If{ξn，n≥1}are nonnegative and WUOD，then for each n≥1

In particular，if{ξn，n≥1}are WUOD，then for each n≥1 and any s＞0

In this paper，we mainly investigate the claim sizes with the heavy-tailed distributions.In the following，some common heavy-tailed distribution classes will be introduced.

For a proper distribution V on（-∞，∞），let V（x）=1-V（x），x∈（-∞，∞）be its tail.Say that a distribution V on（-∞，∞）is a heavy-tailed distribution，if for any λ＞0，；otherwise，say that V is a lighttailed distribution.An important subclass of heavy-tailed distribution class is the dominated varying distribution class，denoted by D.Say that a distribution V on（-∞，∞）belongs to the class D，if for any 0＜y＜1

A related class is the long-tailed distribution class L.Say that a distribution V on（-∞，∞）belongs to the class L，if for any y＞0

Say that a distribution V on[0，∞）belongs to the subexponential distribution class，denoted by V∈S，if

holds for some（or，equivalently，for all）n≥2，where V*ndenotes the n-fold convolution of V.When V is supported on（-∞，∞），we say that V belongs to the subexponential distribution class if V（x）I{x≥0}belongs to this class. It is well known that the following proper inclusion relationships hold

（see，e.g.，[16]）.

In the above risk model，when the inter-arrival times Yn，n≥1，are identically distributed r.v.s and σ=0，Wang et al[12]obtained the following asymptotics of the finite-time ruin probability.Before giving their result，we first introduce a notation of Tang[7].Define Λ={t：λ（t）＞0}with t=inf{t：λ（t）＞0}=inf{t：P（Y1≤t）＞0}.It is clear that Λ=[t，∞]if P（Y1=t）＞0；or Λ=（t，∞]if P（Y1=t）=0.

Theorem1In the above risk model，assume that σ=0，the claims sizes Xn，n≥1，are WUOD r.v.s with common distribution F∈L∩D and the inter-arrival times Yn，n≥1，are identically distributed and WLOD r.v.s. Also suppose that for any ε＞0

We will extend and improve the above result from the following aspects.

1.We will consider a risk model with a perturbed term，i.e.we will consider the case of σ＞0.In practice，the perturbed term can be interpreted as an additional uncertainty of the aggregate claims or the premium income of an insurance company.

2.We will consider a delayed risk model，i.e.in this risk model，Yn，n≥2 are identically distributed，but Y1and Y2are not identically distributed.In this case，N（t），t≥0 is a delayed quasi renewal counting process.

The following is the main result of this paper.

Theorem1In the above risk model，assume that σ≥0，the claims sizes Xn，n≥1，are WUOD r，v，s with common distribution F∈L∩D，the inter-arrival times Yn，n≥1，are WLOD r.v.s and Yn，n≥2，are identically distributed.Also suppose that for any ε＞0，（4）and（5）hold.Then for each finite t∈Λ，the relation（6）holds.

Remark 1①When σ=0，and the claims sizes Xn，n≥1 and the inter-arrival times Yn，n≥1 are i.i.d.r.v.s，respectively，Theorem 2.1 of[8]obtained the relation（6）for the case of F∈S.Theorem 2 extends the result of[8] to a dependent risk model with a perturbed term for the case of F∈L∩D.

②When σ=0，the claims sizes Xn，n≥1 are pairwise NOD，the inter-arrival times Yn，n≥1 are NLOD r.v.s and JF-＞0，[11]obtained the following result under the condition that F∈L∩D：for each finite t∈Λ

Theorem 2 extends the result of[11]to a dependent risk model with a more general dependent structure and with a perturbed term.

2 Proof of result

Before proving the main result，we first give some lemmas.The first lemma extends Lemma 2.1 of[12]to the delayed case.

Lemma1Consider the renewal counting process，n≥1.Suppose that Yn，n≥1 are WLOD r.v.s satisfying（5）and Yn，n≥2 are identically distributed.Also suppose that g（n），n≥1，is a sequence of positive numbers such that for any ε＞0，.Then for any t∈Λ and any v＞0，it holds that

It should be noted that when Yn，n≥1 are identically distributed and g（n）≡1，n≥1 this lemma is Lemma 2.1 of[12].

Proof.We will use the line of the proof of Lemma 2.1 of[12]to prove this lemma.If there exists a positive constant a＞0 such that Y2=a a.s.then Yn，n≥1 are i.i.d..By Lemma 3.2 of[7]，we know that（8）holds.In the follow，we will assume that Y2is not degenerate at any positive constant.Since λ（t）＜∞ for any t≥0，we know that P（Y2=0）＜1.In fact，assume that P（Y2=0）=1，then for any t≥0

Since for any t≥0，λ（t）＜∞，we have for any t≥0，P（Y1≤t）=0，which yields that Y1=∞a.s..This contradicts with the definition of the r.v..By P（Y2=0）＜1 and the proof of（2.2）in[12]，we know that for any t∈Λ there exists T1∈Λ∩[0，T]such that P（Y2≤Y1）＜1.Hence，by（9）-（11）we know that（8）holds.

By Lemma 1 and the proof of Lemma 2.4 of[12]，we can similarly obtain the following result.

Lemma1In the above risk model，suppose that the claim sizes，Xn，n≥1 are WUOD r.v.s with common distribution F∈L∩D，the inter-arrival times，Yn，n≥1 are WLOD r.v.s and Yn，n≥2 are identically distributed. Also suppose that the relations（4）and（5）hold.Then for each finite t∈Λ

By Lemma 2 and F∈L∩D，we know that for each finite t∈Λ，the distribution of Dδ（t）belongs to L∩D.By（12），the dominated convergence theorem and Lemma 2，it holds that for each finite t∈Λ

For the asymptotic upper bound of ψ（x，t），t∈Λ，since，t≥0 is still a Brownian motion，it holds that the distribution ofis light-tailed.Therefore，it follows from Lemma 2 that for each finite t∈Λ

where in the last step，we have used F∈D and the distribution of，t≥0 is light-tailed.Thus，by（6） and Lemmas 1，Lemmas 3 and Lemmas 2，for each finite t∈Λ，we have

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带有扰动项的常利率延迟风险模型的有限时破产概率的渐近估计

高苗苗，王开永*，陈腊梅，钱浩军

（苏州科技大学 数理学院，江苏 苏州 215009）

讨论了带有扰动项的常利率延迟相依风险模型，在此模型中，索赔额与索赔来到时间间隔分别具有某一相依结构，且索赔来到时刻构成一个延迟的准更新记数过程。当索赔额的分布属于某一重尾分布族时，文中得到了上述风险模型的有限时破产概率的渐近性质。

相依风险模型；常利率；有限时破产概率；渐近性质

责任编辑：谢金春

2016-08-04

国家自然科学基金资助项目（11401418）；江苏省“333高层次人才培养工程”资助项目；江苏省自然科学基金资助项目（BK2012165）；苏州科技大学研究生科研创新计划资助项目（SKCX16_056）；江苏省大学生创新创业训练计划资助项目（201510332038Y）

高苗苗（1993-），女，江苏淮安人，硕士研究生，研究方向：概率论及应用。*

王开永（1979-），男，博士，副教授，硕士生导师，E-mail：beewky@vip.163.com。

O211.4MR（2010）Subject Classification：62P05；62E10；60F05

A

：2096-3289（2017）02-0022-06