1 Introduction

Consider a renewal risk model with main and delayed claims in which, for each positive integer i, an insurer’s ith main claim occurs at time accompanied with a delayed claim occurring at time , where denotes an uncertain delay time. Let be a sequence of independent and identically distributed (i.i.d.) non-negative random vectors with generic random vector and marginal distributions F and G, respectively. The accident arrival times constitute a renewal counting process

with a finite mean function , and denote the inter-arrival times by , with , which are i.i.d. nonnegative random variables (r.v.s). The delay times are a sequence of identically distributed and nonnegative (possibly degenerate at 0) r.v.s with common distribution H. The insurer is allowed to invest its surplus into a risk-free market. The price process of the investment portfolio is described by a geometric Lévy process . Here is a nonnegative Lévy process, also representing the stochastic accumulated return rate process, which starts from zero and has independent and stationary increments. For more discussions on Lévy processes, see [1,4,22]. Then the discounted value of the surplus process with stochastic return on investment can be defined as

where 1_A denotes the indicator function of a set is the initial risk reserve of the insurer, is the density function of premium income at time is the volatility factor, , is another nonnegative Lévy process representing the stochastic interest process, and , is the diffusion perturbation, which is a standard Brownian motion. As usual, assume that , , and are mutually independent, but some certain dependence may exist within each pair . Additionally, assume that the premium density function is bounded, i.e., for some and all . For any fixed time , the finite-time ruin probability of risk model (1) can be defined as

and the corresponding infinite-time ruin probability is .

Such a risk model (1) has been playing an important role in insurance practice since a severe accident may trigger more than one claim. First is the main claim caused immediately, while all the others are accumulated as another type, called as the delayed claim, occurring after a stochastic period of time. For example, a traffic accident may cause an immediate payoff for vehicle damage, as well as some medical claims for injures of both drivers and passengers in the subsequent periods. In this paper, we study the asymptotic expression for the finite-time ruin probability, which has immediate implications under modern insurance regulatory frameworks such as solvency capital requirement and insurance risk management.

The ruin probabilities of risk model (1) were initially studied by [28],who considered a Poisson accident-number process and some light-tailed claims and established an exact formula for without perturbation and investment in (1) by a martingale approach. In the presence of heavy-tailed claims, [15] studied the model with two deterministic linear functions for the premium income process and the stochastic accumulated return rate process for premium rate and interest rate ; [16] considered the case of; and both of these two literatures derived the asymptotic relation for . A few extensions with dependence structures and stochastic returns can be found in [9–11,27], who investigated the asymptotic behavior for both and . Some related results in bidimensional risk models can be found in [2,3,26], among others. Remark that all the above works are restricted to some extremely heavy-tailed claims such as the ones with regularly varying tails or consistently varying tails. Recently, by using the asymptotics for the tailprobability of randomly weighted sum of i.i.d.subexponentialr.v.s [25] established an asymptotic formula for the finite-time ruin probability under the independent model (1) with some moderately heavy-tailed (subexponential) claims, but , .

In this paper, we continue to seek the asymptotic behavior for the finite-time ruin probability in a more general risk model (1) with subexponential claims, risk-free investment, and diffusion perturbation, where each pair of the main and delayed claims may be interdependent to some extent. Our obtained results also confirm the intuition that the asymptotic finite-time ruin probability of risk model (1) with subexponential claims is insensitive to the Brownian perturbation, which coincide with the results of the models without delayed claims in [17] and [24]. Our adopting method is the tail asymptotics for the randomly weighted sum of dependent subexponential r.v.s, which may be interesting on its own right.

The rest of this paper consists of four sections. Section 2 states the main result after introducing some necessary preliminaries, and Section 3 performs a simulation study to check the accuracy of the theoretical result. Section 4 establishes some asymptotic formulas for the tail probability of finite randomly weighted sum generated by dependent subexponential r.v.s. The proof of the main result is postponed to Section 5.

2 Preliminaries and main results

Throughout the paper, all limit relationships hold as unless stated otherwise. For two positive functions and , write if lim sup , write if lim , write f if lim , write if lim sup , and write . For two real-valued numbers x and y, denote by and denote the positive and negative parts of x, respectively, by x 0 and 0. The indicator function of a set A is denoted by 1_A. Furthermore, for two positive bivariate functions and, we write uniformly for all t in a nonempty set A if

and write uniformly for all if

Denote by the distribution of a r.v. letting the notation speak for itself. For two real- valued r.v.s and η, we say thatξ is stochastically not greater than η, denoted by , if for all .

2.1 Heavy-tailed distributions

It is well known that most insurance claims possess the heavy-tailed feature since many insurance data are characterised by right heavy-tailedness; see [5,6,21]. We will use heavy-tailed distributions to model the claims. A distribution V on is said to be subexponential, denoted by , if for all and holds for all (or, equivalently, for some) , where is then-fold convolution of V. More generally, a distribution V on is still said to be subexponential if the distribution is subexponential. The class S contains a lot of important distributions such as Pareto, lognormal, and heavy-tailed Weibull distributions. Clearly, by Lemma 1.3.5(a) of [7], if a distribution V on is subexponential, then it holds that

for all , which defines the class of long-tailed distributions denoted by . Automatically, relation (3) holds uniformly on every compact set of y. Hence, it is easy to see that there exists some positive function , with and , such that relation (3) holds uniformly for all |. An important subclass of S is that of regularly varying tailed distributions. A distribution V on is said to be regularly varying tailed with index , denoted by for all . A typical example is the Pareto distribution

with parameters and . The reader is referred to monographs [7] and [8] for reviews of some related heavy-tailed distributions.

2.2 Main results

Our main result is established under the following assumptions, which describe some weak dependence among variables.

Assumption 1. Suppose that real-valued satisfy the relation

for all .

This concept is related to what is called asymptotic independence, see, e.g., [18], and indicates that neither too positively nor too negatively can and be dependent.

Assumption 2. Suppose that for real-valued , there exist two positive constants and M such that

holds for all , and .

When is not a possible value of for some open set ∆ containing , the conditional probability in Assumption 2 is understood as 0. This dependence structure was introduced by [12] and is related to the so-called negative (or positive) regression dependence proposed by [14]. As pointed by [12], if follow a joint n-dimensional Farlie–Gumbel–Morgenstern (FGM) distribution of the form

where are real-valued numbers such that is a proper n-dimensional distribution, are absolutely continuous marginal distributions satisfying then Assumptions 1 and 2 are both satisfied. In addition, Assumption 2 implies Assumption 1.

Now we are ready to state our main result in which Assumption 2 is satisfied for and with and remark that the delay times , can be arbitrarily dependent.

Theorem 1.Consider the risk model (1) in which the generic random vector satisfies Assumption 2 with . Let be any fixed time such that .

(i)If and , then

Where

(ii)If , then

We remark that in Theorem 1, the condition of satisfying Assumption 2 can be reduced to

when for some and . The following corollary is a simplified version of Theorem 1.

Corollary 1. Under the conditions of Theorem 1,assume that for some , the process is a homogeneous Poisson process with intensity , and the distribution H of the delay time is exponentially distributed with intensity . Let be any fixed time such that .

(i) If and , then

Where

(ii)If , then

3 A simulation study

In this section, we use some numerical simulations to verify the accuracy of the asymptotic result for in Corollary 1. To this end, via the crude Monte Carlo (CMC) method we compare the simulated ruin probability in (2) with the asymptotic one on the right-hand side of (6).

Throughout this section, model specifications for the numerical studies are listed below:

• The main and delayed claims X and Y are modelled by a bivariate FGM distribution of (5), which can be reduced to with parameter ; and their marginal distributions are identical to a Pareto distribution (4) with parameters and . Clearly, .

• The accident arrival counting process is a homogeneous Poisson process with intensity . That is, the accident inter-arrival times are i.i.d. nonnegative r.v.s with a common exponential distribution having parameter .

• The delay times are i.i.d. nonnegative r.v.s with a common exponential distribution having parameter .

• The stochastic accumulated return rate process is specialised to

where is a constant, is a homogeneous Poisson process with intensity , and are i.i.d. nonnegative r.v.s; see a similar discussion in [13]. Clearly, such an constitutes a nonnegative Lévy process [4, Prop. 3.10], it can be calculated that in Corollary 1,

Further, assume that is uniformly distributed on [0,1].

• The stochastic interest processreduces to with constant interest rate .

The various parameters are set to:

For the simulated estimation , we first divide the time interval [0,T] into n parts, and for the given , we generate m samples , . Then, for each , generate pairs of , the accident inter-arrival times , and the delay times For each , generate according to (7). Thus, the discounted value of the surplus process can be calculated according to (1). In this way, the ruin probability can be estimated by

In Fig. 1, we compare the CMC estimate in (8) with the asymptotic value given by (6) on the left and show their ratio on the right.

The CMC simulation is conducted with the sample size, the time step size with ,and the initial wealth x from 500 to 3000. From Fig.1 it can be seen that with the increase of the initial wealth x, both estimates decrease gradually and the two lines get closer. In addition, the ratio of the simulated and asymptotic values for the finite-time ruin probability are close to 1. The fluctuation is due to the poor performance of the CMC method, which requires a sufficiently large sample size to meet the demands of high accuracy.

4 Tail behavior of randomly weighted sum

In this section, we investigate the asymptotic tail probability of finite randomly weighted sum generated by dependent subexponential r.v.s, which plays an important role in proving our main result and may be interesting on its own right. Before giving the results, we firstly introduce a series of lemmas. In the sequel, let be a random vector with independent components and the same marginal distributions as those of , which is independent of all the other random sources.

Lemma1.

(i) If

(ii)If

(iii)Let be a real-valued random vector with marginal distributions V₁ and V₂,respectively, and satisfying Assumption 2 with . If and

Proof. The proofs of parts (i) and (ii) are referred to Theorem 3.11 (or Corollary 3.16) and Corollary 3.18 of [8], respectively.

(iii) It is easy to see that if (9) holds. By there exists a functionsuch that , and

holds for any fixed .

On the one hand, for sufficiently large x, according to belonging to , , and, we divide the tail probability into three parts denoted by . By (10) and we have

and

As for , by Assumption 2, for sufficiently large x,

Clearly, part (ii) gives , and similarly to (12), . According to the dominated convergence theorem and , we have . Thus,

Combining (11)–(13), we obtain the upper bound

On the other hand,

It is easy to see that , and further, by Assumption 2 and (10),

Therefore, the lower bound

is derived.

Lemma 2. Let be n real-valued r.v.s with distributions , respectively. If Assumption 1 is satisfied, then for every set , every , and any ,

Proof. Clearly.

which tends to 0 as by Assumption 1.

The next lemma can be derived by using Lemma 2 and the arguments in the proof of Lemma 4.3 of [12]. We omit its detailed proof.

Lemma 3.Let be . real-valued r.v.s with distributions , respectively. If Assumption 1 issatisfied and , then for any and uniformly for all ,

Along the similar line of the proofs of Lemmas 5.1 and 5.2 in [12], we can obtain the following two lemmas in which the uniformity for all , holds by addressing the uniform convergence for the weighted sum with independent subexponential summands and nonrandom weights; see, e.g., Lemma 1 of [23].

Lemma 4.Let be n real-valued r.v.s with distributions _, respectively. If Assumption 2 is satisfied, for some distribution , then there exist two positive numbers x₀ and d_n such that for any and each ,

holds for alland .

Lemma 5. Let be n independent and nonnegative r.v.s with distributions , respectively. If and for some distribution , , then, for any function satisfying and , any , and ,it holds that uniformly for all ,

Lemma 6. Under the conditions of Lemma 4, for ,it holds that uniformly for all ,

Proof. By noting we only prove (14) for nonnegative . Recalling in (10), it holds that

By (10), holds uniformly for all . As for I2, by Lemmas 4 and 5, it holds that for sufficiently large x and uniformly for all ,

Therefore, the desired relation (14) follows.

Combining Lemmas 3 and 6 gives the first result on the uniform asymptotics for the tail probability of the weighted sum with nonrandom weights.

Proposition 1. Under the conditions of Lemma 4, for , it holds that uniformly for all ,

Let be n nonnegative r.v.s satisfying Assumption 2, and be n arbitrarily dependent, nondegenerate at 0,andnonnegative r.v.s independent of .

If are bounded from above, then for each ,

where is the common upper bound of .

By using (15) and Proposition 1 we can mimic the proof of Theorem 1 of [23] to establish the asymptotic formula for the randomly weighted sum of dependent subexponential and nonnegative summands.

Proposition 2.Let be n nonnegative r.v.s with distributions , respectively, and satisfying Assumption 2; and let be n nonnegative r.v.s, which are arbitrarily dependent, bounded from above, nondegenerate at 0, and independent of . If and for some distribution , then

Now we state the last result, which is an extension of Proposition 2.

Proposition 3.Assume that all conditions of Proposition 2 are satisfied. Let η be a real- valued r.v. independent of all other sources. , then

Proof. On the one hand, since is nondegenerate at 0, there exists some small such that . For such ε₀, by the condition we have that for any , there exists some large x₀ such that for all . Construct a new nonnegative r.v. , independent of all other sources, with tail distribution

Clearly, and . Then, by Proposition 2,

by letting then , where in the last step, we used and the fact .

On the other hand, according to Fatou’s lemma and Proposition 2,

where in the last step, we used due to Lemma 2 of [23].

Remark 1. Propositions 2 and 3 still hold if not all weights are degenerate at 0.

5 Proofs of main results

Write the last stochastic integral on the right-hand side of (1) as

and its supremum and infimum as

for any . Before proving the main result, we firstly establish two lemmas. The first one gives the distribution of the above supremum and infimum of the stochastic integral, which may be interesting in its own right.

Lemma 7. Let be a Lévy process, and be a Brownian motion independent of Then, for any fixed and any ,

where Z is a standard normal r.v. independent of

We remark that if for some , then Lemma 7 reduces to Theorem D.3 (ii) of [20].

Proof of Lemma 7. Since is independent of , according to Proposition C.2 of [19], the stochastic integral in (16) is a continuous Ocone martingale; that is, it can be expressed as a time-changed Brownian motion

for some Brownian motion independent of , where is the quadratic variation of . It follows from (18) that

which coincides with (17), where represents equality in distribution.

Further, if is a nonnegative Lévy process, then for any fixed . Hence, by Lemma 7 we have that for any ,

where is the standard Gaussian distribution function. Therefore, by mimicking the proof of Theorem 1.1 of [24] we derive that

Lemma 8.Let be a sequence of i.i.d. nonnegative r.v.s with common distribution belonging to S, let be a renewal counting process with arrival times and mean function , and let and be two nonnegative Lévy processes. Assume that {, are mutually independent. Then, for any such that , it holds that

5.1 Proof of Theorem 1

We firstly prove part (i). We deal with the upper bound for . Choosing some large m, we have that for all ,

For, by Proposition 1, , which implies due to Lemma 1(i), and due to . Then

This, together with Lemma 8, leads to

Now we turn to . For each ,

If , then the first n weights are nondegenerate at 0. Note that by (19) and we have that for all ,

Then, according to Proposition 3 and Remark 1,

Trivially, if , then both sides of (21) are 0,and notation is understood as =. Thus,

Interchanging the order of the sum in yields

As for ,

Applying and Lemma 8 gives that

Combining (20) and (22)–(24), we obtain the upper bound for .

Now we estimate the lower bound for . Since , we have that for any fixed integer m and all ,

Since is nonpositive, we have that for all and any ,

As done in dealing with , we can derive the lower bound for .

The proof of part (ii) is much similar to that of (i), and we only show the difference. For the upper bound, it is easy to see that (20) still holds by using Lemma 1(iii) and the fact

Note that for each ,

Then, as done in (22)–(24), by using Lemma 1(iii), Proposition 3, Lemma 8, and (26) we can obtain

which, together with (20), leads to

For the lower bound, by (25), for any fixed integer m and all ,

As done in (22)–(24), we can derive

5.2 Proof of Corollary 1

We only prove part (i) and omit the similar proof of (ii). By Proposition 3.14 of [4] we have that for any and ,

Since is a homogeneous Poisson process with intensity and the distribution H is exponential with parameter , it can be calculated that ds. Then by and (27) we have

where in the second step, we used the dominated convergence theorem. Similarly,

Therefore, the desired relation follows from Theorem 1.

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