#### Abstract

The insurance risk model involving main claims and by-claims has been traditionally studied under the assumption that every main claim may be accompanied with a by-claim occurring after a period of delay, but in reality each main claim can cause many by-claims arriving according to a counting process. To this end, we construct a new insurance risk model that is also perturbed by diffusion with constant force of interest. In the presence of heavy tails and dependence structures among modelling components, we obtain some asymptotic results for the finite-time ruin probability and the tail probability of discounted aggregate claims, where the results hold uniformly for all times in a finite or infinite interval.

#### 1. Introduction

Consider the classical delay-claims risk model with constant force of interest (see Li [1]), whose wealth process is expressed asHere is the initial capital of an insurance company, is the constant force of interest, is the constant rate of premium, is the indicator function of an event , , are the insurerâ€™s main claims such that each main claim may cause a by-claim , occurring after a delay period , and , are the main-claim arrival times constituting a renewal counting process,with the renewal function for any .

We know that model (1) without interest was introduced by Yuen et al. [2] and then studied by Meng and Wang [3] and Li and Wu [4]. For its various discrete-time counterparts, the reader is referred to Waters and Papatriandafylou [5], Yuen and Guo [6], Xiao and Guo [7], and Wu and Li [8]. All these references above conducted risk analyses with light-tailed and mutually independent claims. But recently, Li [1] considered the model (1) with heavy-tailed claim sizes satisfying a certain dependence structure and gave an asymptotic formula for the ruin probability; Gao et al. [9] extended Liâ€™s result to the case in which the model is perturbed by diffusion and the main-claim inter-arrival times satisfy a certain dependence structure; Gao et al. [10] discussed the uniform asymptotics for the finite-time ruin probability in the delayed-claims risk model with diffusion, dependence structures, and constant force of interest. It is worth mentioning that, in the above-referenced delay-claims risk models, the main claims and the by-claims such that every main claim may be accompanied with only one by-claim occurring after a period of delay were involved. But in insurance practice, for a natural and man-made disaster such as an earthquake or a traffic accident, it is very probable that there are many other insurance claims occurring after the immediate one; namely, each main claim can induce more than one by-claim which practically arrive according to a counting process after the immediate main claim.

In the present paper, we propose a new insurance risk model, which involves the main claims and the by-claims such that each main claim can cause multiple by-claims arriving according to a counting process, and in which all the modelling components satisfy the following assumptions.

*Assumption 1. *Denote by , the interarrival times of main claims with . Assume that the main claims and their interarrival times are two sequences of nonnegative and identically distributed, but not necessarily independent, random variables (r.v.s) with common distributions and , respectively.

*Assumption 2. *For any fixed , we denote, by , the duration from the time when the -th main claim arrives to the time when the -th main claim induces its -th by-claim and by , , the interarrival times of the by-claims caused by the -th main claim with . Then the arrival process of by-claims caused by the -th main claim isnamely, the by-claims caused by the -th main claim arrive according to the above counting process. Assume that the by-claim sizes and their interarrival times are two sequences of nonnegative and identically distributed, but not necessarily independent, r.v.s. with common distributions and , respectively.

*Assumption 3. *The total amount of premiums accumulated up to time , denoted by , is a nonnegative and nondecreasing stochastic process with and almost surely (a.s.) for all .

*Assumption 4. *The diffusion process , as a perturbed term, is a standard Brownian motion with volatility parameter and independent of the other sources of randomness. In practice, the diffusion-perturbed term can be interpreted as an additional uncertainty of the aggregate claims or the premium income of the insurance company.

*Assumption 5. *Assume that and are mutually independent, and so are and .

The delay-claims risk model described by (1) is then extended to a new insurance risk model to match the insurance practice, and the insurerâ€™s total surplus up to time satisfies the following equation:and the discounted aggregate claims up to time can be written asBy Assumption 3, it is easy to see that, for any ,where is the discounted value of premiums accumulated up to time .

We define, as usual, the ruin probability within a finite time byand the infinite-time ruin probability by

All limit relationships in the paper are for unless stated otherwise. For two positive functions and , we write if , write if , write or if , write if and , and write if and .

We notice that, in insurance industry, practitioners usually choose r.v.s. with heavy tails to model large claims. For a distribution and any , we setandwhere and . By definition, we say that a distribution on belongs to the consistently varying-tailed class, denoted by , if belongs to the dominatedly varying-tailed class, denoted by , if, for all , belongs to the long-tailed class, denoted by , if, for all , belongs to the subexponential class, denoted by , if where is the -fold convolution of .

*Remark 6. *Remark that if , then . In fact, for any fixed , is monotonically decreasing function of . So, for , , and then, by , . Because , there exists a sufficiently large number such that for all , and further , . Therefore by the definition of , we know that .

It is well-known thatFor more details of heavy-tailed distributions and their applications to insurance and finance, we refer the readers to Bingham et al. [11], Embrechts et al. [12], and McNeil et al. [13].

In recent years, a study trend of risk theory is to introduce various dependence structures to risk models, among which the widely lower orthant dependence structure was proposed by Wang et al. [14]. Say that r.v.s. are widely lower orthant dependent (WLOD), if there exist a sequence of real numbers such that, for each and all ,Clearly, if are WLOD, then, for each and any ,Chen and Yuen [15] introduced a more general dependence structure, namely, pairwise quasi-asymptotic independence structure. Say that r.v.s. are pairwise quasi-asymptotically independent (PQAI), if, for any ,where and . Adopting the term of Li [1], r.v.s. are said to be pairwise strongly quasi-asymptotically independent (PSQAI), if, for any ,Clearly, if are PSQAI, then they are PQAI. For further study on the above dependence structures and their analogues, we refer to Geluk and Tang [16], Gao and Liu [17], Li [1], Gao et al. [18], and references therein.

Under the above frameworks, in this paper we consider the new insurance risk model (4) with a feature that each main claim can cause multiple by-claims arriving according to a counting process after a period of delay and then investigate the uniform asymptotics for the finite-time ruin probability and the tail probability of the discounted aggregate claims. As generally acknowledged, the finite-time ruin probability is more practical but much harder than the infinite-time ruin probability, and the uniformity considered in the paper often significantly merits the theoretical value of these asymptotic formulas obtained.

The remaining part of this paper is organized as follows: in Section 2 we will state the main results that are proven in Section 4 after giving some lemmas in Section 3.

#### 2. Main Results

In the section, we state the main results of this paper. For notational convenience, we denote by the generic renewal function of the by-claim arrival process , which have the same distribution because of the assumption that are identically distributed in Assumption 2 and the fact that depends only on its inter-arrival times . Besides, for any ,where the second step is due to the result that the process after time has the same distribution as the whole process; i.e., . Define and . Clearly, . With , then

Firstly, we are concerned with the local uniformity of finite-time ruin probability of the risk model (4) with PSQAI sizes of main claims and by-claims arriving according to a sequence of arbitrary counting processes.

Theorem 7. *Consider the risk model introduced by (4) with , where the sizes of main claims and by-claims, are a sequence of PSQAI r.v.s. with common distributions and , respectively, and the arrival processes of main claims and by-claims, and , are a sequence of arbitrary counting processes such that for any fixed , , and , for some . Then, for any fixed and , it holds uniformly for all thatif the premium process is independent of the other sources of randomness.*

*Remark 8. *The uniformity of two bivariate functions for all means that

In comparison to the first theorem, the second one discusses the case when the sizes of main claims and by-claims are PQAI r.v.s. and the premium process is not necessarily independent of the other sources of randomness.

Theorem 9. *Consider the risk model introduced by (4) with , where the sizes of main claims and by-claims, , are a sequence of PQAI r.v.s. with common distributions and , respectively, and the arrival processes of main claims and by-claims, and , satisfy the same conditions as those in Theorem 7. Then relation (22) holds uniformly for all , if, for any fixed ,*

*Remark 10. *Tang [19] pointed out that condition (24), which does not require the independence between the premium process and the other sources of randomness, allows for a more realistic case when the premium income varies as a deterministic or stochastic function of the insurerâ€™s current reserve. The same concerns condition (33) in Theorem 16 below. Indeed, if, for any fixed , for some , then condition (24) is satisfied naturally. By Markovâ€™s inequality,which, along with Lemma 22(2), leads to (24).

*Remark 11. *A comparison of the conditions in Theorems 7 and 9 tells us that the dependence structure among the sizes of main claims and by-claims imposed in Theorem 7 is stronger than that in Theorem 9, but the distribution class of main claims and by-claims in Theorem 7 are larger than that in Theorem 9.

According to Theorems 7 and 9, we now put forward a corollary that gives a uniformly asymptotic formula for the finite-time ruin probability of the risk model (4) with a special case when .

Corollary 12. *Consider the risk model introduced by (4) with ; if the conditions of Theorem 7 or Theorem 9 are valid, then it holds uniformly for all thatwhere is the convolution of the renewal functions and .*

*Remark 13. *Formula (26) contains main-claim distribution and by-claim distribution . Without loss of generality, we now only consider the first relation of (26) to give more explanations on the interrelationship between and as follows: let for some , and one of the conditions below holds.

(1) If , then , which means that if has tail heavier than that of ; then the main claims dominate the asymptotic analysis of the finite-time ruin probability.

(2) If , then ; particularly, if , then , where the second summand of formula (26) inserts only some addition to the first summand.

(3) If , then , which is the opposite of the first case.

*Remark 14. *Considering the conditions on the counting processes and , in Theorems 7 and 9, we observe that the renewal functions and are monotonically nondecreasing and finite for any fixed . Then, it follows that, for all ,which ensures that the second and third expressions of (26) are well-defined. Especially, for the case when the interarrival times of main claims and by-claims, and are two sequences of WLOD r.v.s. such that inequality (32) holds for every , then relation (27) holds for all . In fact, by Markovâ€™s inequality and (17), we get that, for all , Applying (32) and taking for some constant , we see that for some integer such that for all , , which implies thatSimilarly, it also holds thatHence, we obtain relation (27) for all .

*Remark 15. *In Theorems 7 and 9 and Corollary 12, we cannot get the uniformity for all . But in practice, the case for is hardly significant since can be arbitrarily close to .

In the third theorem below, we extend the set over which relation (22) holds uniformly to the maximal set , where the total discounted amount of premiums is assumed to be finite; namely,

Theorem 16. *Under the conditions of Theorem 9 with , we further assume that the interarrival times of main claims and by-claims, and , are two sequences of WLOD r.v.s. such thatholds for every , depending on , , , and . Then relation (22) holds uniformly for all , if one of the following conditions holds.**(1) The premium process is independent of the other sources of randomness.**(2) The total discounted amount of premiums satisfies*

*Remark 17. *Remark that relation (32) as a condition in Theorem 16 must hold for every . See the constants , and their analogous in Section 4, where can be close to , and depend on the distributions and , and and are the arrival times of main claims and by-claims, respectively. Thus, relation (32) must hold for every in order for Theorem 16 to be valid for such arbitrarily given modelling components, and also the constant in (32) depends on the distributions , , , and .

*Remark 18. *By emma 3.3 of Gao et al. [18], the conditions on and in Theorem 16 indicate that, for any fixed , and , , for any , which asserts that the conditions on and , , in Theorems 7 and 9 are more relaxed than those in Theorem 16. In the paper, we consider in Theorems 7 and 9 that the arrival processes of main claims and by-claims are a sequence of arbitrary counting processes, which means that neither independence, nor a special dependence structure, is required among the interarrival times of main claims and by-claims.

By the uniformity of the finite-time ruin probability for all in Theorem 16, we derive the corresponding result on the infinite-time ruin probability (8).

Corollary 19. *Under the conditions of Theorem 16, we have*

Finally, we consider the uniform asymptotics for the tail probability of the discounted aggregate claims, which has the same uniform asymptotics as the finite-time ruin probability in the same time-interval and then can play an important role to prove the first three theorems.

Theorem 20. *Consider the discounted aggregate claims described by (5), if the conditions of Theorem 7 or Theorem 9 are valid; then it holds uniformly for all thatFurthermore, if the conditions of Theorem 16 are valid, relation (35) still holds uniformly for all .*

*Remark 21. *Clearly, if, for each , , namely, every main claim may be accompanied with only one by-claim occurring after a delay period , identically distributed by , then the main results obtained in this paper coincide with those of Gao et al. [10].

#### 3. Some Lemmas

In the section, we prepare some lemmas to prove the main results. The first lemma is due to roposition 2.2.1 of Bingham et al. [11] and emma 3.5 of Tang and Tsitsiashvili [20].

Lemma 22. *If with , then we have the following.**(1) For any , there exist some and such that (2) For any , it holds that .*

The second lemma is from heorem 3.3(iv) of Cline and Samorodnitsky [21] and emma 2.5 of Wang et al. [22].

Lemma 23. *Let be a r.v. with distribution and be a nonnegative r.v., independent of and satisfying for some .**(1) If , then .**(2) If , then the distribution of still belongs to the class .*

The third lemma is a restatement of heorem 2.1 of Li [1]. Also, see emma 3.3 of Gao and Liu [17] and emma 3.2 of Gao et al. [18]. It should be mentioned that the asymptotic formula in this lemma was firstly developed by Tang and Tsitsiashvili [23].

Lemma 24. *If are PSQAI (or PQAI) and real-valued r.v.s. with distributions (or ), , respectively, then, for any fixed ,holds uniformly for all .*

In the following, we present a lemma which plays an important role to prove the main results and is also of its own value.

Lemma 25. *Let be a distribution in the class . If is a r.v. such that and is a nonnegative r.v., independent of and satisfying for some , then*

*Proof. *By , it follows that, for any fixed , there exists such that, for all large ,Hence, for such that , we derive by Markovâ€™s inequality and Lemma 22(2) that, for all large , where is the r.v. distributed by . So by Lemma 23(1) and the arbitrariness of , the lemma holds immediately.

Finally, we give the asymptotic upper-bound of the total discounted amount of aggregate claims as follows.

Lemma 26. *Under the conditions of Theorem 16, it holds that *

*Proof. *Following the proof of emma 3.5 of Gao and Liu [17], we show that there exists a positive integer such thatFor any fixed positive integer , we take , identically distributed by . heorem 3.2 of Chen and Yuen [15] gives thatwhich, along with and Lemma 23(2), leads to . By emma 3.1 of Chen and Yuen [15] (or heorem 2.2 of Li [1]) and heorem 2.5 of Li [1], we obtain that are still PQAI. Then by similar derivation of (42), there exists a positive integer such thatwhere the last step is due to (43) and Lemma 23(1). Similarly to (42), we still show that there exists a positive integer such thatAnd from Lemma 25, we can derive thatIf, for some ,In fact, we arbitrarily choose some positive integer , and set When , we prove by the Cr-inequality and (17) that, for ,Arguing as (29) and setting in (32) for some constant , one easily sees that there exists a positive integer such that, for all , . Hence by (49), we havewhich means that the sequence is bounded from above. Noting that this sequence is monotonically increasing, we apply the monotone bounded convergence theorem to conclude that as , has a finite limit, denoted by , which leads to (47) for . When , by Minkowskiâ€™s inequality and along with the similar lines of the proof of the case when , we also have that, for and for any fixed integer , This, along with the monotone bounded convergence theorem, proves that has a finite limit as , and then (47) still holds for .

Let and be fixed as above. Notice that, for any ,For , by heorem 3.2 of Chen and Yuen [15], we getClearly,andThus,andBecause and , letting yields thatandHence, substituting (58) and (59) into (53) leads to For , combining (42), (44), and (46) and using , and Lemma 23(1), we derive that andTherefore, we substitute the above results of , , into (52) to obtain thatwhere the second term in the last step is due to (20).

#### 4. Proofs of Main Results

In this section, we proceed to prove the main results of this paper. First of all, we should give the proof of Theorem 20 that is helpful to prove Theorems 7â€“16.

*Proof of Theorem 20. *In the first half of this proof, we deal with the uniformity of (35) for all under the conditions of Theorem 7 or Theorem 9, arbitrarily choosing some positive integer . Note that, for all ,Firstly, we consider . For , we denote by and the joint distributions of random vectors and , respectively, where and , and write and . So by Lemma 24, it holds uniformly for all and thatFor , it follows that Clearly, for all , By the condition on , we prove that uniformly for all , as , andFurther, by , we get that uniformly for all , as ,and, similarly, Hence, it holds uniformly for all thatFor , which is divided as by (20), we obtain that, uniformly, for all ,