Abstract

This paper considers a by-claim risk model under the asymptotical independence or asymptotical dependence structure between each main claim and its by-claim. In the presence of heavy-tailed main claims and by-claims, we derive some asymptotic behavior for ruin probabilities.

1. Introduction

Consider a continuous-time risk model with no interest rate in which each severe accident leads to two kinds of claims: one is caused by the accident immediately, called the main claim, and the other is caused as a compensation occurring after a period of time, called the by-claim. The surplus process can be described aswhere stands for the indicator function of a set . In relation (1), is interpreted as the initial reserve of an insurance company, is the constant premium rate, is the size of the th main claim occurring at the arrival time , the counting process represents the number of the accidents (equal to that of the main claims) before time , is the size of the by-claim corresponding to the th main claim, and denotes the uncertain delay time after the accident arrival time . In this setup, the finite-time and infinite-time ruin probabilities can be defined, respectively, as

Such a by-claim risk model is initially studied by Yuen et al. [1] and may be of practical use in insurance. For instance, a traffic accident will cause different kinds of claims, including an immediate payoff for the car damage and another delayed medical claim for injured drivers which may need a random period of time to be settled.

We are concerned with the asymptotic behavior for the finite-time and infinite-time ruin probabilities, since, except for few cases under ideal distributional assumptions, a closed-form expression for the ruin probability or is hardly available. Thus, the mainstream of the study focuses on characterizing the asymptotic behavior for ruin probabilities. Some earlier works have been achieved with light-tailed and mutually independent claims; that is, both and are sequences of independent and identically distributed (i.i.d.) and light-tailed nonnegative random variables, and they are mutually independent too; see Yuen and Guo [2], Xiao and Guo [3], Wu and Li [4], and Li and Wu (2015).

Clearly, modeling total surplus should address extreme risks, which result from the marginal tails of and the tail dependence between claims. In the past decade, more and more attention has been paid to heavy-tailed claims and the dependence between them. In the presence of heavy-tailed claim sizes, Tang [5], Leipus and Šiaulys [6, 7], Yang et al. [8], Wang et al. [9], Liu et al. [10], Yang and Yuen [11], Yang et al. [12, 13], and Chen et al. [14] investigated some independent or dependent risk models with no by-claims. Li [15] considered a dependent by-claim risk model with positive interest rate and extendedly varying tailed main claims and by-claims under the pairwise quasi-asymptotical independence structure (see the definition below). Fu and Li [16] further generalized Li’s result by allowing the insurance company to invest its surplus into a portfolio consisting of risk-free and risky assets. Recently, Li [17] studied a by-claim risk model with no interest rate under the setting that each pair of the main claim and by-claim follows the asymptotical independence structure or possesses a bivariate regularly varying tail (hence, follows the asymptotical dependence structure).

In this paper we continue to consider the above by-claim risk model (1). Precisely speaking, let be a sequence of nonnegative and i.i.d. random vectors, representing the main claims and by-claims, with generic random vector having marginal distributions , and finite means , . Assume that the counting process, not necessarily the renewal one, is generated by the identically distributed and nonnegative interarrival times , with finite mean function . The delay times are a sequence of nonnegative (possibly degenerate at zero) random variables. In addition, we assume that and are mutually independent.

Inspired by the work of Li [17], our goal is to derive sharp asymptotics for the finite-time and infinite-time ruin probabilities in some dependent by-claim risk models. We make some meaningful adjustments and extensions on his model. First, we allow some certain dependence structure among the interarrival times ; that is, is not necessarily a renewal counting process. Second, when investigating the infinite-time ruin probability, we extend both distributions and to be consistently varying tailed in the case that the main claim and the by-claim are asymptotically independent. We also complement another case that and are arbitrarily dependent when dominates .

The rest of the paper is organized as follows. Section 2 prepares some preliminaries including some classes of heavy-tailed distributions and some concepts of dependence structures. Section 3 exhibits our main results. The proofs of the main results as well as several lemmas needed for the proofs are relegated to Section 4.

2. Preliminaries

Throughout the paper, all limit relations are according to unless otherwise stated. For two positive functions and , we write if , write or if , and write if . Furthermore, for two positive bivariate functions and , we write uniformly for if . For any , we write and .

In this paper, we shall restrict the claim distributions to some classes of heavy-tailed distributions supported on . A commonly used class is the class of consistently varying tailed distributions. A distribution belongs to the class , if , where for all . Closely related is a wider class of long-tailed distributions. A distribution belongs to the class , if for any . An important subclass of is the class of regularly varying tailed specified by for any and some , denoted by . The reader is referred to Embrechts et al. [18] and Foss et al. [19] for related discussions on the properties of the subclasses of heavy-tailed distributions.

Modeling a practical risk model must carefully address asymptotical (in)dependence between each pair of the main claim and its corresponding by-claim or among the interarrival times. Asymptotical dependence represents that the probability of two components being simultaneously large cannot be negligible compared with one component being large. Generally, two random variables and are said to be asymptotically dependent if they have a positive coefficient of (upper) tail dependence, defined bysee, e.g., McNeil et al. [20]. The following concept of bivariate regular variation exhibits the asymptotic dependence of two random variables. A random pair taking values in is said to follow a distribution with a bivariate regularly varying (BRV) tail if there exist a distribution and a nondegenerate (i.e., not identically ) limit measure such that the following vague convergence holds: Necessarily, is regularly varying. Assume that for some , for which case we write . By definition, for a random pair , if , then its marginal tails satisfy which imply in (4) showing that and are asymptotically dependent; see, e.g., Tang et al. [21] and Tang and Yang [22]. If in (4) then and are asymptotically independent. A natural extension is the concept of quasi-asymptotical independence (QAI) proposed by Chen and Yuen [23] and defined by In our main results we shall use the structure of BRV or QAI to model the main claim and its corresponding by-claim and capture their tail dependence simultaneously. The following concept of dependence structure is a special case of asymptotical independence, which will be used to describe the inter-arrival times. A sequence of random variables is said to be widely upper orthant dependent (WUOD), if there exists a finite real sequence satisfying, for each and for all ,and it is said to be widely lower orthant dependent (WLOD), if there exists a finite real sequence satisfying, for each and for all ,The sequence of is said to be widely orthant dependent (WOD), if both (8) and (9) hold. Here, are called dominating coefficients. Such dependence structures are introduced by Wang et al. [24]. Specially, when for any in (8) and (9), the sequence of random variables is said to be upper negatively dependent (UND) and lower negatively dependent (LND), respectively. The sequence is said to be negatively dependent (ND), if it is both UND and LND. See Ebrahimi and Ghosh [25]. Note that the ND structure is weaker than the well-known negative association; see Alam and Saxena [26] and Joag-Dev and Proschan [27], among others.

3. Main Results

In this section, we firstly introduce some assumptions on the by-claim risk model (1). Throughout this paper, let be a sequence of i.i.d. random pairs with generic random vector having marginal distributions and both on and finite means and , respectively; let be a sequence of nonnegative and dependent random variables with finite mean , which are independent of ; and let be a sequence of nonnegative and upper bounded random variables; that is, there exists a positive constant such that for all . In addition, as usual in the risk model with no interest rate we require the safety loading condition:We remark that the reasonability for the upper bounded ’s is that by-claims should occur before the termination date due to the insurance policy, and the safety loading condition (10) excludes the trivial case .

The following is our first main result, which investigates the asymptotics for infinite-time ruin probability under three kinds of the dependence structures between and .

Theorem 1. Consider the by-claim risk model (1) with WLOD interarrival times satisfyingfor some . Assume further that either of the following four conditions is satisfied.
Condition 1. is a sequence of LND random variables.
Condition 2. is a sequence of WOD random variables, and there exists a positive and nondecreasing function such that , for some , and ; here, means that there exists some such that Condition 3. is a sequence of WOD random variables with for some , and there exists a constant such that Condition 4. is a sequence of WOD random variables with for some , and, for any , (i)Assume that and are QAI. If and , then(ii)If , for some and , thenwhere .(iii)Assume that and are arbitrarily dependent. If and , then

We remark that if Condition 1 is satisfied, that is, is a sequence of LND random variables, then (11) holds automatically.

Our second result is concerned with the asymptotic behavior of the finite-time ruin probability. Comparing with Theorem 2 of Li [17], we allow to be a quasi-renewal counting process generated by the WOD and identically distributed inter-arrival times , whereas it is required to be a Poisson process in Li [17].

Theorem 2. Under the conditions of Theorem 1, assume that is a sequence of WOD and identically distributed nonnegative random variables such that (11) and Condition 2 in Theorem 1 are satisfied. Then, for every ,holds if either of the following holds: (i) and are QAI, and for some . In this case, .(ii) for some . In this case, .(iii) and are arbitrarily dependent, for some , and . In this case, .

As pointed by Li [17], Theorem 2 shows that, given that the ruin occurs, the ruin time divided by converges in distribution to a Pareto random variable of type II. In addition, if is a sequence of ND random variables, then (11) and Condition 2 in Theorem 1 are both satisfied automatically.

4. Proofs of Main Results

Before the proofs of our two main results, we firstly cite a series of lemmas. Consider a nonstandard risk modelwhere is a sequence of i.i.d. nonnegative random variables with generic random variable . As (2) and (3), define the infinite-time and finite-time ruin probabilities of risk model (19) with replaced by , denoting them by and , respectively. The following first lemma comes from Corollary of Wang et al. [9]; see some similar results in Yang et al. [8].

Lemma 3. Consider a nonstandard risk model (19), in which the claims form a sequence of i.i.d. nonnegative random variables with common distribution and finite mean ; the interarrival times are a sequence of nonnegative, NLOD, and identically distributed random variables with finite positive mean such that (11) is satisfied; and and are mutually independent. Assume further that either of Conditions 1–4 in Theorem 1 is satisfied. If and (10) holds, then, for any such that ,holds uniformly for all , where .

We remark that in Lemma 3 the uniformity of (20) implies

We point out that Corollary 2.1 of Wang et al. [9] gave some more situations when discussing the uniform asymptotics for , in which they allowed the distribution to be strongly subexponential.

The second lemma can be found in Yang et al. [12].

Lemma 4. Let be a random vector with marginal distributions and both on , respectively, but and are arbitrarily dependent. If and , then

The third lemma gives the elementary renewal theorem for WOD random variables, which is due to Theorem 1.4 of Wang and Cheng [28].

Lemma 5. Let be a sequence of nonnegative, WOD, and identically distributed random variables with finite positive mean , which constitutes a quasi-renewal counting process with mean function . If Condition 2 in Theorem 1 is satisfied, then as .

As pointed out by Tang [5], it is easy to see that, under the conditions of Lemma 3, by using Lemma 5, relation (20) holds with replaced by , but the range of the uniformity for becomes smaller.

Lemma 6. Under the conditions of Lemma 3, if, in addition, is a sequence of nonnegative, WOD, and identically distributed random variables with finite positive mean , such that (11) and Condition 2 in Theorem 1 are satisfied, then holds uniformly for all , where is an arbitrary infinitely increasing function and .

Proof. By Lemmas 3 and 5, for any and all , when is sufficiently large, we haveFor the upper bound, by the right-hand side of (24) we have that for sufficiently large and all , which concludes the upper bound by the arbitrariness of . We next consider the lower bound. Similarly, by the left-hand side of (24) we have that for sufficiently large and all , for some , where the last step holds because, by and , This completes the proof of the lemma.

We are now ready for the proofs of the two main results.

Proof of Theorem 1. We firstly consider the tail probability of . In case (i), since and are QAI and , , by using Theorem 3.1 of Chen and Yuen [23] we haveimplying . In case (ii), by we haveimplying . In case (iii), applying Lemma 4 givesalso implying . The above three relations, together with (21), lead toBy the first equivalence of (31), we have for any fixed because of . Then, we can follow the same line of Li [17] to verifyTherefore, the desired relations (15)–(17) hold from (31) and (32).

Proof of Theorem 2. Under the conditions of Theorem 2, by (28)–(30) we have with specified in Theorem 2. By Lemma 6 we have that, for every , by this and (31), according to Karamata’s theorem, we deriveSimilarly to (32), Li [17] provedThus, the desired relation (18) follows from (32), (34), and (35). This completes the proof of the theorem.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by National Natural Science Foundation of China (NSFC: 71671166).