Approximation for the Finite-Time Ruin Probability of a General Risk Model with Constant Interest Rate and Extended Negatively Dependent Heavy-Tailed Claims
We propose a general continuous-time risk model with a constant interest rate. In this model, claims arrive according to an arbitrary counting process, while their sizes have dominantly varying tails and fulfill an extended negative dependence structure. We obtain an asymptotic formula for the finite-time ruin probability, which extends a corresponding result of Wang (2008).
1. The Dependent General Risk Model
In this paper, we consider the finite-time ruin probability with constant interest rate in a dependent general risk model. In this model, the claim sizes form a sequence of identically distributed, not necessarily independent, and nonnegative random variables (r.v.s) with common distribution such that for all ; the claim arrival process is a general counting process, namely, a nonnegative, nondecreasing, right continuous, and integer-valued stochastic process with for all large . The times of the successive claims are denoted by . The total amount of premiums accumulated up to time , denoted by with and almost surely for every , is another nonnegative and nondecreasing stochastic process. Assume that , and are mutually independent. Let be the constant interest rate (i.e., after time one dollar becomes dollars), and let be the initial capital reserve of an insurance company. Then, the total discounted reserve up to time , denoted by , can be written as For a finite time , the finite-time ruin probability is defined by while the ultimate ruin probability is defined by
If the claim sizes are independent r.v.s, the model is called the independent general risk model, which was introduced by Wang . In particular, if , , with a deterministic constant and is a Poisson process, then the model reduces to the classical one.
2. Introduction and Main Result
Hereafter, all limit relationships hold for tending to unless otherwise stated. For two positive functions and , we write if ; write if and if . The indicator function of an event is denoted by .
In risk theory, heavy-tailed distributions are often used to model large claim amounts. They play a key role in insurance and finance. We will restrict the claim-size distribution to be heavy tailed. A distribution is said to be dominatedly varying tailed, denoted by , if for any . A distribution is said to be long tailed, denoted by , if for any . A distribution is said to be subexponential, denoted by , if for any , where denotes the -fold convolution of itself. A distribution is said to be regularly varying tailed, denoted by , if for any . A proper inclusion relationship holds that see, for example, Cline  or Embrechts and Omey . For a distribution , denote the upper Matuszewska index of the distribution by In the terminology of Bingham et al. , the quantity is actually the upper Matuszewska index of the function , , as also pointed out in Tang and Tsitsiashvili . Additionally, denote (clearly, ). The presented definitions yield that the following assertions are equivalent:
The asymptotic behavior of the ruin probability in the classical risk model has been extensively investigated in the literature. Klüppelberg and Stadtmüller  considered the ultimate ruin probability for the case of regularly-varying-tailed claim sizes. Using the reflected random walk theory, Asmussen  extended the study to a larger class of heavy-tailed distributions; see Corollary 4.1(ii) of his paper. Complementary discussions on the ultimate ruin probability can be found in Kalashnikov and Konstantinides , Konstantinides et al. , Tang , among others.
In this paper, we are interested in the finite-time ruin probability. In this aspect, Tang  established an asymptotic result in the classical risk model: under the condition , he obtained that for every for which , Recently, Wang  derived some important and interesting results in two independent risk models. One is the delayed renewal risk model, in which (2.4) holds if ; another is the general risk model, in which (2.4) also holds if . We are interested in the latter, for example, the general risk model, and restate Theorem 2.2 of Wang  here.
Theorem 2.1. In the independent general risk model introduced in Section 1, assume that the claim sizes are independent and identically distributed nonnegative r.v.s with common distribution . Assume that for any with , there exists some constant such that Then, (2.4) holds.
We call r.v.s are extended negatively dependent (END) if there exists some positive constant such that both hold for each and all . This dependence structure was introduced by Liu . Recall that r.v.s are called upper negatively dependent (UND) if (2.6) holds with , they are called lower negatively dependent (LND) if (2.7) holds with , and they are called negatively dependent (ND) if both (2.6) and (2.7) hold with . These negative dependence structures were introduced by Ebrahimi and Ghosh  and Block et al. . Clearly, ND r.v.s must be END r.v.s., and Example 4.1 of Liu  shows that the END structure also includes some other dependence structures.
Motivated by the work of Wang , under the END structure, we formulate our main result as follows.
Theorem 2.2. In the dependent general risk model introduced in Section 1, assume that the claim sizes are END nonnegative r.v.s with common distribution and finite mean . Assume that for any with , there exists some constant such that Then, it holds that Furthermore, if , then (2.4) holds.
The rest of the present paper consists of two sections. We give some lemmas and the proof of Theorem 2.2 in Section 3. In Section 4, we perform some numerical calculations to verify the approximate relationship in our main result.
3. Proof of Main Result and Some Lemmas
In the sequel, and always represent some finite and positive constants whose values may vary in different places. In this section, we start by giving some lemmas to show some properties of the class and the END structure. The first lemma is a combination of Proposition 2.2.1 of Bingham et al.  and Lemma 3.5 of Tang and Tsitsiashvili .
Lemma 3.1. If a distribution , then (i)for any , there exist positive constants and such that holds for all and(ii) it holds for every that .
By direct verification, END r.v.s have the following properties similar to those of ND r.v.s; see Lemma 3.1 of Liu . For some refined properties of END r.v.s, one can refer to Chen et al. . The following lemma can also be found in Lemma 2.2 of Chen et al. .
(i) If r.v.s are nonnegative and END, then for any , there exists some positive constant such that
(ii) If r.v.s are END and are either all monotone increasing or all monotone decreasing, then are still END.
The following two lemmas play important roles in the proof of our main result.
Lemma 3.3. Let be identically distributed and END r.v.s with common distribution and . Then, for any , and , there exists some positive constant such that
Proof. Following the proof of Lemma 2.3 of Tang , we employ a standard truncation argument to prove this lemma. For simplicity, we write , . If , then is almost surely nonpositive for each , implying for any positive , and thus (3.1) holds.
Let, in the following, . For any fixed and positive integer , define According to Lemma 3.2(ii), and are still END r.v.s, respectively. Denote , . Clearly, It remains to estimate the second summand in (3.3). For a positive , by Lemma 3.2(ii), are END nonnegative r.v.s. Hence, using identity by Markov inequality and Lemma 3.2(i) we have Since for all and is strictly increasing in , from (3.5), we obtain Choose , which is positive. For such , by (3.6), we have The last estimate and (3.3) imply the desired estimate (3.1). The lemma is proved.
Lemma 3.4. In the dependent general risk model introduced in Section 1, assume that the claim sizes are END nonnegative r.v.s with common distribution . Let be an arbitrary nonnegative r.v. and assume that , and are mutually independent. Then, for any and any positive integer , Furthermore, if , then
We remark that if is consistently varying tailed (see the definition in Chen and Yuen ), then by conditioning (3.9) easily follows from Theorem 3.2 of Chen and Yuen . Note that this case is in a broader scope, since there is no need to assume independence between and .
Proof. We follow the line of the proof of Lemma 3.6 of Wang  with some modifications in relation to the properties of the class and the END structure. Clearly, for each ,
We first show the upper bound. For any fixed ,
By , for any and each , we have uniformly for all and ,
by firstly letting then . We note that are END r.v.s. Then, by , there exists some positive constant such that for sufficiently large , each , all and ,
Since can be arbitrarily large, it follows that
Hence, from (3.10)–(3.14), we obtain for each ,
As for the lower bound for (3.10), since are END r.v.s, we have for sufficiently large and each ,
holds uniformly for all and . By and Fatou's lemma, we have for any and all ,
Similar to (3.15), from (3.10), (3.16), and (3.18), we obtain for each ,
The desired relation (3.8) follows now from (3.15) and (3.19).
If , (3.9) follows by using the properties of the class to establish analogies of relations (3.12) and (3.17). This ends the proof of the lemma.
Proof of Theorem 2.2. We use the idea in the proof of Theorem 2.2 of Wang  (e.g., Theorem 2.1 of this paper) to prove this result. Clearly, and imply . By (2.8), we have for any , there exists some positive integer such that
To estimate the upper bound of , we split it into two parts as According to Lemma 3.4 of this paper and Lemma 3.5 of Wang , we have for sufficiently large , By Lemma 3.3, , Lemma 3.1(ii), (3.20), and , there exists some positive constant such that for sufficiently large , By Lemma 3.1(i), for any , there exists some positive constant such that for sufficiently large , which, combining (3.23) and , implies From (3.21), (3.22), and (3.25), we derive the right-hand side of (2.9).
As for the lower bound of , by Lemma 3.4, we have for the above given and sufficiently large , Analogously to the estimate for , we have for sufficiently large , From (3.26) and (3.27), we obtain the left-hand side of (2.9).
If , then (2.4) follows by using (3.9) in the proof of (3.22) and (3.26).
4. Numerical Calculations
In this section, we perform some numerical calculations to check the accuracy of the asymptotic relations obtained in Theorem 2.2. The main work is to estimate the finite-time ruin probability defined in (1.2).
We assume that the claim sizes come from the common Pareto distribution with parameter , , which belongs to the class , and are independent replications of with the joint distribution with parameter , where the joint distribution is constructed according to the Frank Copula. It has been proved in Example 4.2 of Liu  that and are END r.v.s. Since are independent copies of , the r.v.s are END as well.
Assume that the claim arrival process is the homogeneous Poisson process with intensity parameter . Clearly, such an integer-valued process satisfies the condition (2.8). Choose . The total amount of premiums is simplified as with the premium rate , and the constant interest rate . Here, we set the time as and the initial capital reserve , respectively. We aim to verify the accuracy of relation (2.4). The procedure of the computation of the finite-time ruin probability in Theorem 2.2 is listed here.
Step 1. Assign a value for the variable and set .
Step 2. Divide the close interval into pieces, and denote each time point as , .
Step 3. For each , generate a random number from the Poisson distribution , and set as the sample size of the claims.
Step 5. Calculate the expression below for each and denote them as : where and have been defined and their values have also been assigned.
Step 6. Select the maximum value from , and denote it as , compare with ; if , then the value of increases 1.
Step 8. Calculate the moment estimate of the finite-time ruin probability, .
For different value of , the analog value and the theoretical result of the finite-time ruin probability are presented in Table 1, and the percentage of the error relative to the theoretical result is also presented in the bracket behind the analog value. It can be found that from Table 1, the larger becomes, the smaller the difference between the analog value and the theoretical result is. Therefore, the approximate relationship in Theorem 2.2 is reasonable.
The authors would like to thank the two referees for their useful comments on an earlier version of this paper. The revision of this work was finished during a research visit of the first author to Vilnius University. He would like to thank the Faculty of Mathematics and Informatics for its excellent hospitality. Research supported by National Natural Science Foundation of China (no. 11001052), China Postdoctoral Science Foundation (20100471365), National Science Foundation of Jiangsu Province of China (no. BK2010480), Natural Science Foundation of the Jiangsu Higher Education Institutions of China (no. 09KJD110003), Postdoctoral Research Program of Jiangsu Province of China (no. 0901029C), and Jiangsu Government Scholarship for Overseas Studies, Qing Lan Project.
N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge, UK, 1987.
Q. Tang and G. Tsitsiashvili, “Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks,” Stochastic Processes and their Applications, vol. 108, no. 2, pp. 299–325, 2003.View at: Publisher Site | Google Scholar | Zentralblatt MATH
D. Konstantinides, Q. Tang, and G. Tsitsiashvili, “Estimates for the ruin probability in the classical risk model with constant interest force in the presence of heavy tails,” Insurance: Mathematics and Economics, vol. 31, no. 3, pp. 447–460, 2002.View at: Publisher Site | Google Scholar | Zentralblatt MATH