Asymptotics of finite-time ruin probability of a risk model with dependent main-claims and by-claims

GUO Xiaojuan, WANG Kaiyong

(School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou 215009, Jiangsu, China)

Abstract： A risk model with main-claims and by-claims is considered in the paper. The risk model with a constant force of interest and a Brownian perturbation are studied. When the main-claims and by-claims are pairwise quasi-asymptotically independent with dominatedly varying tails, the asymptotics of finite-time ruin probability have been obtained．

Key words： finite-time ruin probability; main-claims; by-claims; asymptotics

1 Introduction

1.1 Risk model

(1)

And the discounted aggregate claims up to timet≥0 is

(2)

where 1Ais the indicator function of an eventA. The ruin probability within a finite timet>0 is defined as

ψr(x,t)=P(Ur(s)<0 for some 0≤s≤t|Ur(0)=x).

(3)

This paper will investigate the asymptotics of the finite-time ruin probabilityψr(x,t) fort>0asx→∞. In this paper, we assume that {Xi,Yi,i≥1},{θi,i≥1},{Ti,i≥1},{C(t),t≥0}and {B(t),t≥0} are independent and for any 0

(4)

At the end of this subsection, we state some notions. If there is no special statement, all limit relationships in this paper are forx→∞. For two nonnegative functionsa(x) andb(x), we writea(x)=o(1)b(x)if limsupa(x)/b(x)=0; writea(x)b(x)if limsupa(x)/b(x)≤1; writea(x)b(x) if liminfa(x)/b(x)≥1; writea(x)～b(x) if lima(x)/b(x)=1. Letx∧y=min{x,y}andx∨y=max{x,y}. For a proper distributionVon (-∞,∞), letthe tail ofVforx∈(-∞,∞). As Tang[1], we defineΛ={t:E(N(t))>0}={t:P(τ1≤t)>0}andClearly,

1.2 Main results

This paper will consider the main claims and by-claims are heavy-tailed. In the following we will introduce some heavy-tailed distribution classes. For detailed definitions and properties of these distribution classes, one can see Bingham et al[2], Embrechts et al[3]and so on．

Say that a distributionVon (-∞,∞) belongs to the heavy-tailed class, denoted byV∈K, if for anys>0,

Say that a distributionVon (-∞,∞)belongs to the long-tailed class, denoted byV∈S, if for anyy>0,

holds for some (or equivalently for all)n=2,3,…, whereV*ndenotes then-fold convolution ofVwith itself. Say that a distributionVon (-∞,∞) belongs to the classS, ifV(x)1{x≥0}∈S. Say that a distributionVon (-∞,∞) belongs to the dominatedly-varying-tailed class, denoted byV∈D, if for any 0

It is well known that

L∩D?S?L．

For a distributionVon (-∞,∞), we define that

For the risk model (1), Fu et al[4]supposed that the main-claims and by-claims are a sequence of dependent random variables with dominatedly varying tails, asymptotics of the ruin probability were investigated. Fu and Li[5]supposed that the surplus was invested to a portfolio of one risk-free asset and one risky asset with dependent main-claims and by-claims. For the above result, Li[6]considered other dependence structures in which each pair random variables were quasi-asymptotically independent or had the structure of bivariate regular variation. Under the condition that the distributions of main-claims and by-claims belonged to the regularly-varying-tailed distribution class, an asymptotic estimation of the infinite-time ruin probability was obtained. Gao and Liu[7]constructed a new insurance risk model that was also perturbed by diffusion with a constant force of interest which involved main claims and by-claims. They obtained some asymptotic results for the finite-time ruin probability and the tail probability of discounted aggregate claims. Gao et al[8]considered a risk model with a constant force of interest under the PSQAI structure, which is introduced by Li[9]. Say that the real random variables {ξi,i≥1} are pairwise strong quasi-asymptotically independent (PSQAI), if for any 1≤i≠j<∞ and allxi,xj∈(-∞,∞),

In this paper, we mainly discuss the above delayed-claim risk model for PSQAI claims with dominatedly-varying-tailed distributions. The following is the main result of this paper．

(5)

2 Proof of main result

Before giving the proof of Theorem 1, we first present some lemmas. The first is from Proposition 2.2.1 of Bingham et al[2]．

(6)

and

The following Lemma 2 is from Lemma 2 of Li et al[10]．

Lemma2Letξbe a r.v. with a distributionVon [0,∞) andηbe a nonnegative r.v., which is independent ofξ. LetKbe the distribution ofξη. IfV∈DthenK∈D．

Lemma3Let {ξi,i≥1}and{ηi,i≥1} be two sequences of nonnegative and identically distributed r.v.s with distributionsV∈DandK∈D, respectively. If {ξi,ηi,i≥1} are PSQAI then for any fixedn≥1 and 0

(7)

ProofThe result of the left side of (7) is Lemma 2.4 (1) of Wang et al[11]. Now we prove the right side of (7). For any 00,

J1(x)+J2(x).

Hence, by the proof of Lemma 2.4 of Wang et al[11], it holds uniformly for (c1,c2,…,cn)∈[a,b]nthat

Thus,

13.Will you give me your youngest daughter?: Here we have one of the first motifs57 which make this tale very similar to Beauty and the Beast. A beast asks for the youngest, beautiful daughter. The implication is that he wants to marry her, although a wedding ceremony is usually not acknowledged or detailed until the end of the tale once the enchantment58 has been broken.Return to place in story.

Similarly, we can obtain that

Hence, it holds uniformly for (c1,c2,…,cn)∈[a,b]nand (d1,d2,…,dn)∈[a,b]nthat

The following Lemma 4 is from Lemma 3.4 of Gao et al[8]．

Lemma4Consider the risk model (1). It holds for allt∈Λthat

P(Yie-r(τi+Ti)1{τi+Ti≤t}>x))=

For the risk model (1), Lemma 3.6 of Gao et al[8]gave an upper bound of the tail of the discounted aggregate claimsDr(t),t≥0 forF∈L∩DandG∈L∩D. The following lemma considers the case thatF∈DandG∈D．

Lemma5Under the conditions of Theorem 1, for any fixedt0∈Λ∩(0,∞)andT∈(t0,∞), it holds uniformly for allt∈[t0,T] that

(8)

and

(9)

ProofWe use the way of Tang[12]to prove this lemma. For any fixed positive integermandx>0, it holds that

P(Dr(t)>x)=

B1(x)+B2(x).

(10)

DenoteK(t1,…,tn+1) andV(u1,…,un) be the joint distributions of random vectors (τ1,…,τn+1) and(T1,…,Tn),1≤n≤m, respectively. Write Ωn,t={0≤t1≤…≤tn≤t,tn+1>t}. Thus, by Lemma 3, it holds uniformly for allt∈[t0,T] and 1≤n≤mthat

Thus, it holds uniformly for allt∈[t0,T] that

B1(x)

L-1(B11(x)-B12(x)).

(11)

ForB11(x), it holds uniformly for allt∈[t0,T] that

(12)

ForB12(x), sinceF∈DandG∈D, it holds that

(13)

Thus by (11)-(13) and Lemma 4, it holds uniformly for allt∈[t0,T] that

(14)

(15)

Hence, by (14),F∈D,G∈DandE(N(T))P+1<∞, it holds uniformly for allt∈[t0,T]that

(16)

Therefore, by (10)-(12) and Lemma 4 we get that (8) holds. By the proof of (8), we can get that (9) holds.

The last lemma is from Lemma 2.2 of Wang and Mao[13]．

(17)

and

(18)

SinceF∈DandG∈D, by Lemma 5 we getDr(t)∈D,t>0 and it holds uniformly for allt∈[t0,T] that

(19)

Thus, by (17), (19) and Lemma 6, it holds for any fixedt∈Λthat

(20)

(21)

LetM↑∞ in (21), it holds for any fixedt∈Λthat

LWP(Dr(t)>x).

(22)

Thus, by (17), (22) and Lemma 5, it holds for any fixedt∈Λthat

ψr(x,t)LWP(Dr(t)>x)

(23)

By (21) and (23), we get that (5) holds.