Uniform asymptotics for the tail of the discounted aggregate claims with NQD claim sizes

2021-09-22 09:33:30CUIYongfangWANGKaiyong

蘇州科技大學學報(自然科學版) 2021年3期

CUI Yongfang，WANG Kaiyong

（School of Mathematical Sciences，SUST，Suzhou 215009，China）

Abstract：This paper considers a risk model with stochastic return，in which the price process of the investment portfolio is expressed as a geometric Lévy process.When the claim sizes have consistently varied distributions，the uniform asymptotics of the tail of the discounted aggregate claims has been obtained for the negatively quadrant dependent claim sizes，which extends some corresponding results of this risk model.

Key words：uniform asymptotics；discounted aggregate claims；Lévy process；heavy-tailed distributions

1 Introduction and main result

This paper considers a risk model，where the claim size{Xn，n≥1}is a sequence of identically distributed random variable（r.v.）with common distributionFand the inter-arrival time{θn，n≥1}is a sequence of independent r.v.and whose members have a common distributionG.

which represents the number of claims up to timet.Suppose its mean function λ（t）=EN（t）＜∞for allt≥0 and λ（t）→∞ast→∞.In this model，the insurer can make risk-free and risky investments.We use a geometric Lévy processwith a Lévy process{Rt，t≥0}to describe the price process of the investment portfolio，where the Lévy process{Rt，t≥0}starts from 0 and owns independent and stationary increments.Suppose{Xn，n≥1}，{θn，n≥1}and{Rt，t≥0}are mutually independent.Assume that the Lévy process{Rt，t≥0}is right continuous with left limit.LetE[R1]＞0，thenRtdrifts to∞almost surely ast→∞.Define the Laplace exponent for the Lévy process{Rt，t≥0}to be φ（z）=logE[e-zR1]，z∈（-∞，∞）.If φ（z）is finite thenE[e-zRt]=etφ（z）＜∞for anyt≥0（see Proposition 3.14 of[1]）.

The discounted aggregate claims up to timet≥0 is denoted by

whereIAis the indicator function of eventA.As[2]，we define

Lett=inf{t:λ（t）＞0}=inf{t:P（θ1≤t）＞0}.IfP（θ1=t）＞0，then Λ=[t，∞]；ifP（θ1=t）＝0，then Λ=（t，∞].In order to simplify the investigation，we make an assumption thatt＝0.Set Λε=[ε，∞]for any ε＞0.

For the asymptotics of the tail of the discounted aggregate claims，when the processRt=rt，t≥0，r＞0，there are many papers to deal with the independent or dependent claim sizes，such as Gao et al[3]，Chen et al[4]，Yang et al[5]，Wang et al[6]，Hao and Tang[7]，Tang[2，8].For the general Lévy processRt，t≥0，Tang et al[9]considered the independent claim sizes and inter-arrival times.Li[10]and Yang et al[11]investigated a time-dependent risk model，where{（Xn，θn），n≥1}is assumed to be a sequence of independent and identically distributed（i.i.d.）random vector.Yang et al[12]studied the asymptotics of the tail of the discounted aggregate claims for the dependent claim sizes with extended regularly varying tails.In this paper，we still consider this dependent risk model and discuss the asymptotics of the tail of aggregate for claim sizes belonging to a larger distribution class than before.

In this paper，all limit relationships are forx→∞unless stated otherwise.For two positive functionsu（·）andv（·），denoteu（x）v（x）orv（x）u（x）if lim supu（x）/v（x）≤1；writeu（x）~v（x）if limu（x）/v（x）=1；if lim supu（x）/v（x）＜∞，then denoteu（x）=O（v（x））；if limu（x）/v（x）=0 writeu（x）=o（v（x））.Leta∨b=max{a，b}anda∧b=min{a，b}.For a distributionVon（-∞，∞），letV（x）=1-V（x），x∈（-∞，∞）be its tail.

In the following，we will introduce some heavy-tailed distribution subclasses which are used in this paper.

We say that a distributionVon（-∞，∞）belongs to the dominated varying distribution class which is denoted byV∈D，if for any 0＜y＜1，

Say that a distributionVon（-∞，∞）belongs to the consistently varying distribution class which is denoted byV∈C，if

Say that a distributionVon（-∞，∞）belongs to the extended-regularly-varying distribution class，if there exist some constants 0＜α≤β＜∞such that for anyy≥1，

In this case we denoteV∈ERV（-α，-β）.By the definition，the following relation holds

（see，e.g.[13]）.

For a distributionVon（-∞，∞），we denote its upper and lower Matuszewska indices respectively by

（see，e.g.[14]）.

This paper mainly investigates dependent claim sizes.We will give an example——the subprime crisis in 2006 in the United States.People found that heavy tail could better describe the distribution of claims than light tail in that crisis.When claims caused by financial products which were guaranteed by insurance companies for large banks occurred，claims were not mutually independent and there was a certain dependence among those claims.So we will introduce some dependence structures.Let{ξn，n≥1}be a sequence of r.v.{ξn，n≥1}is called to be pairwise negatively quadrant dependent（NQD）if for anyi≠j≥1 and anyx，y∈（-∞，∞），

（see，e.g.[15]）.

Now，we present the main result of this paper.

Theorem 1 For the discounted aggregate claims（1），suppose the claim size{Xn，n≥1}is a sequence of NQD r.v.with common distribution F∈C and JF-＞0.If there exists a constantp＞JF+such that φ（p）＜0，then

hold uniformly for allt∈Λ ifP（θ1=0）＞0；and uniformly for allt∈ΛεifP（θ1=0）=0.

Remark 1 Theorem 2.2 of[12]obtained the uniform asymptotics of the discounted aggregate claims（1）forF∈ERV（-α，-β），0＜α≤β＜∞.Theorem 1 extends Theorem 2.2 of[12]for the distribution of claims sizes fromF∈ERV（-α，-β），0＜α≤β＜∞toF∈C.

2 Proof of main result

2.1 Some lemmas

We will show relation（2）by proving

and

hold uniformly for allt∈Λ ifP（θ1=0）＞0；and uniformly for allt∈ΛεifP（θ1=0）=0，respectively.In the proof of the first relation Lemma 6 and Lemma 7 are used and in the proof of the second relation Lemma 4，Lemma 6 and Lemma 7 are used.Lemma 1-Lemma 5 are use in the proofs of Lemma 6 and Lemma 7.

Lemma 1（Proposition 2.21 of[14]）IfV∈Dthen，

（ⅰ）for any 0＜p1＜JV-≤JV+＜p2andc＞1，there exists some constantD＞0 such that

hold forxy＞Dandx＞D；

（ⅱ）for anyp＞JV+，

The following lemma is the result of Lemma 3.2 of[11]for the independent case between the claim size{Xn，n≥1}and inter-arrival time{θn，n≥1}.

Lemma 2 Under the conditions of Theorem 1，for anyk≥1，it hold uniformly for allt∈Λ that

The next two lemmas are special cases of Lemma 3.3 and Lemma 3.5 of[11].

Lemma 3 Under the conditions of Theorem 1，there is some positive functiona（x）satisfyinga（x）→∞anda（x）=o（x）such that，for anyk≥1，it hold uniformly for allt∈Λ that

Lemma 4 Under the conditions of Theorem 1，for anyk≥1 and any fixed 0＜y＜1，

hold uniformly for allt∈Λ ifP（θ1=0）＞0；and uniformly for allt∈ΛεifP（θ1=0）=0.

By a proof similar to that of Lemma 3.6 of[10]，the following lemma can be obtained.

Lemma 5 Under the conditions of Theorem 1，for anyj＞k≥1，it hold uniformly for allt∈Λ that

Lemma 6 Under the conditions of Theorem 1，for anyn≥1，

hold uniformly for allt∈Λ ifP（θ1=0）＞0；and uniformly for allt∈ΛεifP（θ1=0）=0.

Proof We use the same argument as that for Lemma 3.7 of[10].On one hand，for any 0＜ε＜1，we have

By Lemma 4，

hold uniformly for allt∈Λ ifP（θ1=0）＞0；and uniformly for allt∈ΛεifP（θ1=0）=0.

ForI2（x，n，t），by Lemma 5 and Lemma 4 and the arbitrariness of ε，

hold uniformly for allt∈Λ ifP（θ1=0）＞0；and uniformly for allt∈ΛεifP（θ1=0）=0.Combining（6）-（8）with the arbitrariness of ε and F∈C，we obtain that

hold uniformly for allt∈Λ ifP（θ1=0）＞0；and uniformly for allt∈ΛεifP（θ1=0）=0．

On the other hand，it is clear that

hold uniformly for allt∈Λ，where we use Lemma 5 in the third step.So we have that

hold uniformly for allt∈Λ.This completes the proof of this lemma. □

Lemma 7 Under the conditions of Theorem 1，it holds that

and

Proof We follow the line of proof of Lemma 3.8 of[10]to prove this lemma.Since 0＜JF-≤JF+＜∞，we may take 0＜p1＜JF-andJF+＜p2＜p.By（3），there exists some constantD1＞0 such that，forx＞D1andxy＞D1，