Approximation for the Finite-Time Ruin Probability of a General Risk Model with Constant Interest Rate and Extended Negatively Dependent Heavy-Tailed Claims

Abstract

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 {𝑋_𝑛,𝑛≥1} form a sequence of identically distributed, not necessarily independent, and nonnegative random variables (r.v.s) with common distribution 𝐹 such that 𝐹(𝑥)=1−𝐹(𝑥)=P(𝑋₁>𝑥)>0 for all 𝑥>0; the claim arrival process {𝑁(𝑡),𝑡≥0} is a general counting process, namely, a nonnegative, nondecreasing, right continuous, and integer-valued stochastic process with 0<E𝑁(𝑡)=𝜆(𝑡)<∞ for all large 𝑡>0. The times of the successive claims are denoted by {𝜏_𝑛,𝑛≥1}. The total amount of premiums accumulated up to time 𝑡≥0, denoted by 𝐶(𝑡) with 𝐶(0)=0 and 𝐶(𝑡)<∞ almost surely for every 𝑡>0, is another nonnegative and nondecreasing stochastic process. Assume that {𝑋_𝑛,𝑛≥1}, {𝑁(𝑡),𝑡≥0} and {𝐶(𝑡),𝑡≥0} are mutually independent. Let 𝛿>0 be the constant interest rate (i.e., after time 𝑡 one dollar becomes 𝑒^𝛿𝑡 dollars), and let 𝑥≥0 be the initial capital reserve of an insurance company. Then, the total discounted reserve up to time 𝑡≥0, denoted by 𝐷(𝑡,𝑥), can be written as 𝐷(𝑡,𝑥)=𝑥+^𝑡0𝑒^−𝛿𝑠𝐶(d𝑠)−^𝑁(𝑡)_𝑛=1𝑋_𝑛𝑒^{−𝛿𝜏𝑛}.(1.1) For a finite time 𝑇>0, the finite-time ruin probability is defined by Ψ(𝑥,𝑇)==PP(𝐷s(u𝑡,p𝑥_{𝑡∈[0,𝑇]})<0,forsome0≤𝑡≤𝑇)^𝑁(𝑡)_𝑛=1𝑋_𝑛𝑒^{−𝛿𝜏𝑛}−^𝑡0𝑒^−𝛿𝑠𝐶(d𝑠)>𝑥,(1.2) while the ultimate ruin probability is defined by Ψ(𝑥)=Ψ(𝑥,∞)=P(𝐷(𝑡,𝑥)<0,forsome𝑡≥0).(1.3)

If the claim sizes {𝑋_𝑛,𝑛≥1} are independent r.v.s, the model is called the independent general risk model, which was introduced by Wang [1]. In particular, if 𝐶(𝑡)=𝑐𝑡, 𝑡≥0, with 𝑐>0 a deterministic constant and {𝑁(𝑡),𝑡≥0} is a Poisson process, then the model reduces to the classical one.

2. Introduction and Main Result

Hereafter, all limit relationships hold for 𝑥 tending to ∞ unless otherwise stated. For two positive functions 𝑓(𝑥) and 𝑔(𝑥), we write 𝑓(𝑥)∼𝑔(𝑥) if lim𝑓(𝑥)/𝑔(𝑥)=1; write 𝑓(𝑥)≲𝑔(𝑥) if limsup𝑓(𝑥)/𝑔(𝑥)≤1 and 𝑓(𝑥)=𝑜(𝑔(𝑥)) if lim𝑓(𝑥)/𝑔(𝑥)=0. The indicator function of an event 𝐴 is denoted by 𝟏_𝐴.

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 𝐹 to be heavy tailed. A distribution 𝑉 is said to be dominatedly varying tailed, denoted by 𝑉∈𝒟, if limsup𝑉(𝑥𝑦)/𝑉(𝑥)<∞ for any 𝑦>0. A distribution 𝑉 is said to be long tailed, denoted by 𝑉∈ℒ, if lim𝑉(𝑥+𝑦)/𝑉(𝑥)=1 for any 𝑦>0. A distribution 𝑉 is said to be subexponential, denoted by 𝑉∈𝒮, if 𝑉^𝑛∗(𝑥)∼𝑛𝑉(𝑥) for any 𝑛≥2, where 𝑉^𝑛∗ denotes the 𝑛-fold convolution of itself. A distribution 𝑉 is said to be regularly varying tailed, denoted by ℛ_−𝛼,𝛼>0, if lim𝑉(𝑥𝑦)/𝑉(𝑥)=𝑦^−𝛼 for any 𝑦≥1. A proper inclusion relationship holds that ℛ_−𝛼⊂ℒ∩𝒟⊂𝒮⊂ℒ,(2.1) see, for example, Cline [2] or Embrechts and Omey [3]. For a distribution 𝑉, denote the upper Matuszewska index of the distribution 𝑉 by 𝐽^+𝑉=−lim_𝑦→∞log𝑉_∗(𝑦)log𝑦with𝑉_∗(𝑦)=liminf_𝑥→∞𝑉(𝑥𝑦)𝑉(𝑥),𝑦>1.(2.2) In the terminology of Bingham et al. [4], the quantity 𝐽^+𝑉 is actually the upper Matuszewska index of the function 1/𝑉(𝑥), 𝑥≥0, as also pointed out in Tang and Tsitsiashvili [5]. Additionally, denote 𝐿_𝑉=lim_𝑦↘1𝑉_∗(𝑦) (clearly, 0≤𝐿_𝑉≤1 ). The presented definitions yield that the following assertions are equivalent: (i)𝑉∈𝒟,(ii)𝑉_∗(𝑦)>0forsome𝑦>1,(iii)𝐿_𝑉>0,(iv)𝐽^+𝑉<∞.(2.3)

The asymptotic behavior of the ruin probability in the classical risk model has been extensively investigated in the literature. Klüppelberg and Stadtmüller [6] considered the ultimate ruin probability for the case of regularly-varying-tailed claim sizes. Using the reflected random walk theory, Asmussen [7] 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 [8], Konstantinides et al. [9], Tang [10], among others.

In this paper, we are interested in the finite-time ruin probability. In this aspect, Tang [11] established an asymptotic result in the classical risk model: under the condition 𝐹∈𝒮, he obtained that for every 𝑇>0 for which 𝜆(𝑇)>0, Ψ(𝑥,𝑇)∼^𝑇0−𝐹𝑥𝑒^𝛿𝑡𝜆(d𝑡).(2.4) Recently, Wang [1] derived some important and interesting results in two independent risk models. One is the delayed renewal risk model, in which (2.4) holds if 𝐹∈𝒮; another is the general risk model, in which (2.4) also holds if 𝐹∈ℒ∩𝒟. We are interested in the latter, for example, the general risk model, and restate Theorem   2.2 of Wang [1] here.

Theorem 2.1. In the independent general risk model introduced in Section 1, assume that the claim sizes {𝑋_𝑛,𝑛≥1} are independent and identically distributed nonnegative r.v.s with common distribution 𝐹∈ℒ∩𝒟. Assume that for any 𝑇>0 with 𝜆(𝑇)−𝜆(0)>0, there exists some constant 𝜂=𝜂(𝑇)>0 such that E(1+𝜂)^𝑁(𝑇)<∞.(2.5) Then, (2.4) holds.

In the present paper, we aim to deal with the extended negatively dependent general risk model to get a similar result under 𝐹∈𝒟. Simultaneously, the condition (2.5) can be weakened to (2.8) below.

We call r.v.s {𝜉_𝑛,𝑛≥1} are extended negatively dependent (END) if there exists some positive constant 𝑀 such that both P^𝑛_𝑘=1𝜉_𝑘>𝑦_𝑘≤𝑀^𝑛_𝑘=1PP𝜉_𝑘>𝑦_𝑘,(2.6)_𝑛_𝑘=1𝜉_𝑘≤𝑦_𝑘≤𝑀^𝑛_𝑘=1P𝜉_𝑘≤𝑦_𝑘(2.7) hold for each 𝑛≥1 and all 𝑦₁,…,𝑦_𝑛. This dependence structure was introduced by Liu [12]. Recall that r.v.s {𝜉_𝑛,𝑛≥1} are called upper negatively dependent (UND) if (2.6) holds with 𝑀=1, they are called lower negatively dependent (LND) if (2.7) holds with 𝑀=1, and they are called negatively dependent (ND) if both (2.6) and (2.7) hold with 𝑀=1. These negative dependence structures were introduced by Ebrahimi and Ghosh [13] and Block et al. [14]. Clearly, ND r.v.s must be END r.v.s., and Example  4.1 of Liu [12] shows that the END structure also includes some other dependence structures.

Motivated by the work of Wang [1], 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 {𝑋_𝑛,𝑛≥1} are END nonnegative r.v.s with common distribution 𝐹∈𝒟 and finite mean 𝜇. Assume that for any 𝑇>0 with 𝜆(𝑇)−𝜆(0)>0, there exists some constant 𝑝>𝐽^+𝐹 such that E(𝑁(𝑇))^𝑝<∞.(2.8) Then, it holds that 𝐿_𝐹^𝑇0−𝐹𝑥𝑒^𝛿𝑡𝜆(d𝑡)≲Ψ(𝑥,𝑇)≲𝐿^𝐹−1^𝑇0−𝐹𝑥𝑒^𝛿𝑡𝜆(d𝑡).(2.9) Furthermore, if 𝐹∈ℒ∩𝒟, 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, 𝑀 and 𝑎 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 𝒟 and the END structure. The first lemma is a combination of Proposition  2.2.1 of Bingham et al. [4] and Lemma  3.5 of Tang and Tsitsiashvili [15].

Lemma 3.1. If a distribution 𝑉∈𝒟, then (i)for any 𝛾>𝐽^+𝑉, there exist positive constants 𝑎 and 𝑏 such that 𝑉(𝑦)/𝑉(𝑥)≤𝑎(𝑦/𝑥)^−𝛾 holds for all 𝑥≥𝑦≥𝑏 and(ii) it holds for every 𝛾>𝐽^+𝑉 that 𝑥^−𝛾=𝑜(𝑉(𝑥)).

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 [12]. For some refined properties of END r.v.s, one can refer to Chen et al. [16]. The following lemma can also be found in Lemma  2.2 of Chen et al. [16].

Lemma 3.2. (i) If r.v.s {𝜉_𝑛,𝑛≥1} are nonnegative and END, then for any 𝑛≥1, there exists some positive constant 𝑀 such that E(∏^𝑛𝑘=1𝜉_𝑘)≤𝑀∏^𝑛𝑘=1E𝜉_𝑘.
(ii) If r.v.s {𝜉_𝑛,𝑛≥1} are END and {𝑓_𝑛(⋅),𝑛≥1} are either all monotone increasing or all monotone decreasing, then {𝑓_𝑛(𝜉_𝑛),𝑛≥1} are still END.

The following two lemmas play important roles in the proof of our main result.

Lemma 3.3. Let {𝜉_𝑛,𝑛≥1} be identically distributed and END r.v.s with common distribution 𝑉 and 𝜇^+𝑉=E𝜉₁𝟏_{𝜉1≥0}<∞. Then, for any 𝜃>0, 𝑥>0 and 𝑛≥1, there exists some positive constant 𝑀 such that P^𝑛_𝑘=1𝜉_𝑘>𝑥≤𝑛𝑉(𝜃𝑥)+𝑀𝑒𝜇^+𝑉𝑛𝑥^𝜃−1.(3.1)

Proof. Following the proof of Lemma  2.3 of Tang [17], we employ a standard truncation argument to prove this lemma. For simplicity, we write 𝑆^𝜉𝑛=∑^𝑛𝑘=1𝜉_𝑘, 𝑛≥1. If 𝜇^+𝑉=0, then 𝜉_𝑛 is almost surely nonpositive for each 𝑛≥1, implying P(𝑆^𝜉𝑛>𝑥)=0 for any positive 𝑥, and thus (3.1) holds.
Let, in the following, 𝜇^+𝑉>0. For any fixed 𝜃>0 and positive integer 𝑛, define 𝜉𝜉̃̃_𝑛=min𝜉_𝑛,𝜃𝑥_+𝑛,=max𝜉̃_𝑛,0=𝜉_𝑛𝟏_{{0≤𝜉𝑛≤𝜃𝑥}}+𝜃𝑥𝟏_{{𝜉𝑛>𝜃𝑥}}.(3.2) According to Lemma 3.2(ii), {𝜉̃_𝑛,𝑛≥1} and {𝜉̃^+𝑛,𝑛≥1} are still END r.v.s, respectively. Denote 𝑆^𝜉𝑛=∑^𝑛𝑘=1𝜉̃_𝑘, 𝑛≥1. Clearly, P𝑆^𝜉𝑛>𝑥=≤𝑛P𝑆^𝜉𝑛>𝑥,max_{1≤𝑘≤𝑛}𝜉_𝑘>𝜃𝑥+P𝑆^𝜉𝑛>𝑥,max_{1≤𝑘≤𝑛}𝜉_𝑘≤𝜃𝑥𝑉(𝜃𝑥)+P𝑆^𝜉𝑛>𝑥.(3.3) It remains to estimate the second summand in (3.3). For a positive ℎ, by Lemma 3.2(ii), {𝑒^ℎ𝜉̃^+𝑛,𝑛≥1} are END nonnegative r.v.s. Hence, using identity E𝑒^ℎ𝜉̃⁺¹=^0𝜃𝑥𝑒^ℎ𝑢−1𝑉(d𝑢)+𝑒^ℎ𝜃𝑥−1𝑉(𝜃𝑥)+1,(3.4) by Markov inequality and Lemma 3.2(i) we have P𝑆^𝜉𝑛>𝑥≤𝑒^−ℎ𝑥E𝑒^ℎ𝑆^𝜉𝑛≤𝑒^−ℎ𝑥E𝑒^{ℎ∑𝑛𝑘=1}𝜉̃^+𝑘≤𝑒^−ℎ𝑥𝑀(E𝑒^ℎ𝜉̃⁺¹)^𝑛=𝑀𝑒^−ℎ𝑥^0𝜃𝑥𝑒^ℎ𝑢−1𝑉(d𝑢)+𝑒^ℎ𝜃𝑥−1𝑉(𝜃𝑥)+1^𝑛.(3.5) Since 1+𝑢≤𝑒^𝑢 for all 𝑢∈ℝ and (𝑒^ℎ𝑢−1)/𝑢 is strictly increasing in 𝑢>0, from (3.5), we obtain P𝑆^𝜉𝑛>𝑥≤𝑀exp𝑛^0𝜃𝑥𝑒^ℎ𝑢−1𝑢𝑢𝑉(d𝑢)+𝑛𝑒^ℎ𝜃𝑥≤𝑀exp𝑛𝑒_ℎ𝜃𝑥−1𝑉(𝜃𝑥)−ℎ𝑥−1𝜃𝑥^0𝜃𝑥≤𝑀exp𝑛𝑒_ℎ𝜃𝑥𝑢𝑉(d𝑢)+𝜃𝑥𝑉(𝜃𝑥)−ℎ𝑥−1𝜃𝑥𝜇^+𝑉−ℎ𝑥.(3.6) Choose ℎ=(𝜃𝑥)⁻¹log(𝑥(𝜇^+𝑉𝑛)⁻¹+1), which is positive. For such ℎ, by (3.6), we have P𝑆^𝜉𝑛>𝑥≤𝑀exp1≤𝑀exp𝜃1−𝜃1log𝜇𝑥^+𝑉𝑛+1𝜃log𝑒𝜇^+𝑉𝑛𝑥.(3.7) 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 {𝑋_𝑛,𝑛≥1} are END nonnegative r.v.s with common distribution 𝐹∈𝒟. Let 𝑍 be an arbitrary nonnegative r.v. and assume that {𝑋_𝑛,𝑛≥1}, {𝑁(𝑡),𝑡≥0} and 𝑍 are mutually independent. Then, for any 𝑇>0 and any positive integer 𝑛₀, 𝐿_𝐹𝑛⁰_𝑘𝑘=1_𝑗=1P𝑋_𝑗𝑒^{−𝛿𝜏𝑗}>𝑥,𝑁(𝑇)=𝑘≲^𝑛0_𝑘=1P^𝑘_𝑗=1≲𝐿_{𝐹𝑛−10}𝑋_𝑗𝑒^{−𝛿𝜏𝑗}>𝑥+𝑍,𝑁(𝑇)=𝑘_𝑘𝑘=1_𝑗=1P𝑋_𝑗𝑒^{−𝛿𝜏𝑗}>𝑥,𝑁(𝑇)=𝑘.(3.8) Furthermore, if 𝐹∈ℒ∩𝒟, then ^𝑛0_𝑘=1P^𝑘_𝑗=1𝑋_𝑗𝑒^{−𝛿𝜏𝑗}>𝑥+𝑍,𝑁(𝑇)=𝑘∼^𝑛0_𝑘𝑘=1_𝑗=1P𝑋_𝑗𝑒^{−𝛿𝜏𝑗}>𝑥,𝑁(𝑇)=𝑘.(3.9)

We remark that if 𝐹 is consistently varying tailed (see the definition in Chen and Yuen [18]), then by conditioning (3.9) easily follows from Theorem  3.2 of Chen and Yuen [18]. Note that this case is in a broader scope, since there is no need to assume independence between (𝜏₁,…,𝜏_𝑛0) and 𝑍.

Proof. We follow the line of the proof of Lemma  3.6 of Wang [1] with some modifications in relation to the properties of the class 𝒟 and the END structure. Clearly, for each 𝑘=1,…,𝑛₀, P^𝑘_𝑗=1=𝑋_𝑗𝑒^{−𝛿𝜏𝑗}>𝑥+𝑍,𝑁(𝑇)=𝑘_{{0≤𝑡1≤⋯≤𝑡𝑘≤𝑇,𝑡𝑘+1>𝑇}}^∞0−P^𝑘_𝑗=1𝑋_𝑗𝑒^{−𝛿𝑡𝑗}>𝑥+𝑧×P𝑍∈d𝑧,𝜏₁∈d𝑡₁,…,𝜏_𝑘+1∈d𝑡_𝑘+1.(3.10) We first show the upper bound. For any fixed 𝑙>0, P^𝑘_𝑗=1𝑋_𝑗𝑒^{−𝛿𝑡𝑗}>𝑥+𝑧≤P^𝑘_𝑗=1𝑋_𝑗+P_𝑘𝑒^{−𝛿𝑡𝑗}>𝑥+𝑧−𝑙_𝑗=1𝑋_𝑗𝑒^{−𝛿𝑡𝑗}>𝑥+𝑧,max_{1≤𝑗≤𝑘}𝑋_𝑗𝑒^{−𝛿𝑡𝑗}≤𝑥+𝑧−𝑙∶=𝐼₁+𝐼₂.(3.11) By 𝐹∈𝒟, for any 0<𝜃<1 and each 𝑘=1,…,𝑛₀, we have uniformly for all 𝑡₁,…,𝑡_𝑘∈[0,𝑇] and 𝑧∈[0,∞), 𝐼₁≤^𝑘_𝑗=1𝐹𝜃(𝑥+𝑧)𝑒^𝛿𝑡𝑗≲𝐿^𝐹𝑘−1_𝑗=1𝐹(𝑥+𝑧)𝑒^𝛿𝑡𝑗,(3.12) by firstly letting 𝑥→∞ then 𝜃↗1. We note that {𝑋_𝑛,𝑛≥1} are END r.v.s. Then, by 𝐹∈𝒟, there exists some positive constant 𝑀=𝑀(𝑛₀) such that for sufficiently large 𝑥, each 𝑘=1,…,𝑛₀, all 𝑡₁,…,𝑡_𝑘∈[0,𝑇] and 𝑧∈[0,∞), 𝐼₂=P^𝑘_𝑗=1𝑋_𝑗𝑒^{−𝛿𝑡𝑗}>𝑥+𝑧,𝑥+𝑧𝑘<max^{1≤𝑗≤𝑘}≤P_𝑘𝑋_𝑗𝑒^{−𝛿𝑡𝑗}≤𝑥+𝑧−𝑙_𝑖=1_𝑗≠𝑖𝑋_𝑗𝑒^{−𝛿𝑡𝑗}>𝑙,𝑋_𝑖𝑒^{−𝛿𝑡𝑖}>𝑥+𝑧≤𝑘_𝑘_𝑖=1_𝑗≠𝑖P𝑋_𝑗𝑒^{−𝛿𝑡𝑗}>𝑙𝑘−1,𝑋^𝑖𝑒^{−𝛿𝑡𝑖}>𝑥+𝑧≤𝑀_𝑘𝑘_𝑖=1_𝑗≠𝑖𝐹𝑙𝑒^𝛿𝑡𝑗𝑘−1𝐹(𝑥+𝑧)𝑒^𝛿𝑡𝑖≤𝑀𝑘𝐹𝑙𝑛₀−1^𝑘_𝑗=1𝐹(𝑥+𝑧)𝑒^𝛿𝑡𝑗.(3.13) Since 𝑙 can be arbitrarily large, it follows that limsup_𝑙→∞limsup_𝑥→∞sup_{𝑡1,…,𝑡𝑘∈[][0,𝑇,𝑧∈0,∞)}𝐼₂∑^𝑘𝑗=1𝐹(𝑥+𝑧)𝑒^𝛿𝑡𝑗=0.(3.14) Hence, from (3.10)–(3.14), we obtain for each 𝑘=1,…,𝑛₀, P^𝑘_𝑗=1𝑋_𝑗𝑒^{−𝛿𝜏𝑗}>𝑥+𝑍,𝑁(𝑇)=𝑘≲𝐿^𝐹𝑘−1_𝑗=1_{{0≤𝑡1≤⋯≤𝑡𝑘≤𝑇,𝑡𝑘+1>𝑇}}^∞0−×P𝑍∈d𝑧,𝜏₁𝐹(𝑥+𝑧)𝑒^𝛿𝑡𝑗=𝐿_𝐹𝑘−1∈d𝑡₁,…,𝜏_𝑘+1∈d𝑡_𝑘+1_𝑗=1≤𝐿_𝐹𝑘−1P𝑋_𝑗𝑒^{−𝛿𝜏𝑗}>𝑥+𝑍,𝑁(𝑇)=𝑘_𝑗=1P𝑋_𝑗𝑒^{−𝛿𝜏𝑗}>𝑥,𝑁(𝑇)=𝑘.(3.15) As for the lower bound for (3.10), since {𝑋_𝑛,𝑛≥1} are END r.v.s, we have for sufficiently large 𝑥 and each 𝑘=1,…,𝑛₀, P^𝑘_𝑗=1𝑋_𝑗𝑒^{−𝛿𝑡𝑗}>𝑥+𝑧≥P^𝑘_𝑗=1𝑋_𝑗≥𝑒^{−𝛿𝑡𝑗}>𝑥+𝑧_𝑘_𝑗=1≥𝐹(𝑥+𝑧)𝑒^𝛿𝑡𝑗−_{1≤𝑖<𝑗≤𝑘}P𝑋_𝑖𝑒^{−𝛿𝑡𝑖}>𝑥+𝑧,𝑋_𝑗𝑒^{−𝛿𝑡𝑗}>𝑥+𝑧_𝑘_𝑗=1=(1−𝐹𝑜((1𝑥))+𝑧_𝑘)𝑒^𝛿𝑡𝑗−𝑀_{1≤𝑖<𝑗≤𝑘}𝐹(𝑥+𝑧)𝑒^𝛿𝑡𝑖𝐹(𝑥+𝑧)𝑒^𝛿𝑡𝑗_𝑗=1𝐹(𝑥+𝑧)𝑒^𝑟𝑡𝑗(3.16) holds uniformly for all 𝑡₁,…,𝑡_𝑘∈[0,𝑇] and 𝑧∈[0,∞). By 𝐹∈𝒟 and Fatou's lemma, we have for any 𝜃̃>1 and all 𝑗=1,2,…, liminf1𝐹(𝑥)P𝑋^𝑗>𝑥+𝑍𝑒^𝛿𝑇=liminf^∞0−𝐹𝑥+𝑧𝑒^𝛿𝑇≥𝐹(𝑥)P(𝑍∈d𝑧)_∞0−liminf𝐹𝜃̃𝑥=𝐹(𝑥)P(𝑍∈d𝑧)𝐹_∗𝜃̃⟶𝐿_𝐹,𝜃̃↘1,(3.17) which means P𝑋_𝑗>𝑥+𝑍𝑒^𝛿𝑇≳𝐿_𝐹𝐹(𝑥).(3.18) Similar to (3.15), from (3.10), (3.16), and (3.18), we obtain for each 𝑘=1,…,𝑛₀, P^𝑘_𝑗=1≳𝑋_𝑗𝑒^{−𝛿𝜏𝑗}>𝑥+𝑍,𝑁(𝑇)=𝑘_𝑘_𝑗=1≥P𝑋_𝑗𝑒^{−𝛿𝜏𝑗}>𝑥+𝑍,𝑁(𝑇)=𝑘_𝑘_𝑗=1_{{0≤𝑡1≤…≤𝑡𝑘≤𝑇,𝑡𝑘+1>𝑇}}≳𝐿_𝐹𝑘P𝑋_𝑗>𝑥𝑒^𝛿𝑡𝑗+𝑍𝑒^𝛿𝑇P𝜏₁∈d𝑡₁,…,𝜏_𝑘+1∈d𝑡_𝑘+1_𝑗=1_{{0≤𝑡1≤…≤𝑡𝑘≤𝑇,𝑡𝑘+1>𝑇}}=𝐿_𝐹𝑘𝐹𝑥𝑒^𝛿𝑡𝑗P𝜏₁∈d𝑡₁,…,𝜏_𝑘+1∈d𝑡_𝑘+1_𝑗=1P𝑋_𝑗𝑒^{−𝛿𝜏𝑗}>𝑥,𝑁(𝑇)=𝑘.(3.19) The desired relation (3.8) follows now from (3.15) and (3.19).
If 𝐹∈ℒ∩𝒟, (3.9) follows by using the properties of the class ℒ 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 [1] (e.g., Theorem 2.1 of this paper) to prove this result. Clearly, 𝐹∈𝒟 and 𝜇<∞ imply 𝐽^+𝐹≥1. By (2.8), we have for any 𝜖>0, there exists some positive integer 𝑛₁=𝑛₁(𝑇,𝜖) such that E(𝑁(𝑇))^𝑝𝟏_{{𝑁(𝑇)>𝑛1}}≤𝜖.(3.20)
To estimate the upper bound of Ψ(𝑥,𝑇), we split it into two parts as Ψ(𝑥,𝑇)≤P^𝑁(𝑇)_𝑗=1=⎛⎜⎜⎝𝑋_𝑗𝑒^{−𝛿𝜏𝑗}>𝑥_𝑛1_𝑘=1+^∞_𝑘=_𝑛1+1⎞⎟⎟⎠P^𝑘_𝑗=1𝑋_𝑗𝑒^{−𝛿𝜏𝑗}>𝑥,𝑁(𝑇)=𝑘∶=𝐼₃+𝐼₄.(3.21) According to Lemma 3.4 of this paper and Lemma  3.5 of Wang [1], we have for sufficiently large 𝑥, 𝐼₃≤(1+𝜖)𝐿^{𝐹𝑛−11}_𝑘𝑘=1_𝑗=1≤(1+𝜖)𝐿_𝐹∞−1P𝑋_𝑗𝑒^{−𝛿𝜏𝑗}>𝑥,𝑁(𝑇)=𝑘_𝑗=1=(1+𝜖)𝐿_𝐹∞−1P𝑋_𝑗𝑒^{−𝛿𝜏𝑗}>𝑥,𝑁(𝑇)≥𝑗_𝑗=1=(1+𝜖)𝐿_𝐹−1P𝑋_𝑗𝑒^{−𝛿𝜏𝑗}>𝑥,𝜏_𝑗≤𝑇^𝑇0−𝐹𝑥𝑒^𝛿𝑡𝜆(d𝑡).(3.22) By Lemma 3.3, 𝐹∈𝒟, Lemma 3.1(ii), (3.20), and 𝑝>𝐽^+𝐹≥1, there exists some positive constant 𝑀 such that for sufficiently large 𝑥, 𝐼₄≤^∞_𝑘=_𝑛1+1P^𝑘_𝑗=1≤𝑋_𝑗>𝑥P(𝑁(𝑇)=𝑘)𝐹𝑝⁻¹𝑥^∞_𝑘=_𝑛1+1𝑘P(𝑁(𝑇)=𝑘)+𝑀(𝑒𝜇)^𝑝𝑥^∞−𝑝_𝑘=_𝑛1+1≤𝑀𝑘^𝑝P(𝑁(𝑇)=𝑘)=𝑀𝜖𝐹(𝑥)E𝑁(𝑇)𝟏_{{𝑁(𝑇)>𝑛1}}+E(𝑁(𝑡))^𝑝𝟏_{{𝑁(𝑇)>𝑛1}}𝐹(𝑥).(3.23) By Lemma 3.1(i), for any 𝛾>𝐽^+𝐹, there exists some positive constant 𝑎 such that for sufficiently large 𝑥, ^𝑇0−𝐹𝑥𝑒^𝛿𝑡𝜆(d𝑡)≥≥𝑎𝑎⁻¹₋₁𝐹(𝑥)^𝑇0−𝑒^{−𝛾𝛿𝑡}𝜆(d𝑡)𝑒^{−𝛾𝛿𝑇}(𝜆(𝑇)−𝜆(0))𝐹(𝑥),(3.24) which, combining (3.23) and 𝜆(𝑇)−𝜆(0)>0, implies 𝐼₄≤𝑀𝜖^𝑇0−𝐹𝑥𝑒^𝛿𝑡𝜆(d𝑡).(3.25) From (3.21), (3.22), and (3.25), we derive the right-hand side of (2.9).
As for the lower bound of Ψ(𝑥,𝑇), by Lemma 3.4, we have for the above given 𝜖>0 and sufficiently large 𝑥, Ψ(𝑥,𝑇)≥P^𝑁(𝑇)_𝑗=1≥𝑋_𝑗𝑒^{−𝛿𝜏𝑗}>𝑥+^𝑇0𝑒^−𝛿𝑠𝐶(d𝑠)_𝑛1_𝑘=1P^𝑘_𝑗=1≥(1−𝜖)𝐿𝑋_{𝑗𝐹𝑛1}𝑒^{−𝛿𝜏𝑗}>𝑥+^𝑇0𝑒^−𝛿𝑠𝐶(d𝑠),𝑁(𝑇)=𝑘_𝑘𝑘=1_𝑗=1=(1−𝜖)𝐿_𝐹P𝑋_𝑗𝑒^{−𝛿𝜏𝑗}>𝑥,𝑁(𝑇)=𝑘⎛⎜⎜⎝^∞_𝑗=1P𝑋_𝑗𝑒^{−𝛿𝜏𝑗}>𝑥,𝜏_𝑗≤𝑇−^∞_𝑘=_𝑛1𝑘+1_𝑗=1∶=(1−𝜖)𝐿_𝐹P𝑋_𝑗𝑒^{−𝛿𝜏𝑗}>𝑥,𝑁(𝑇)=𝑘⎞⎟⎟⎠^𝑇0−𝐹𝑥𝑒^𝛿𝑡𝜆(d𝑡)−𝐼₅.(3.26) Analogously to the estimate for 𝐼₄, we have for sufficiently large 𝑥, 𝐼₅≤𝐹(𝑥)E𝑁(𝑇)𝟏_{{𝑁(𝑇)>𝑛1}}≤𝑀𝜖^𝑇0−𝐹𝑥𝑒^𝛿𝑡𝜆(d𝑡).(3.27) From (3.26) and (3.27), we obtain the left-hand side of (2.9).
If 𝐹∈ℒ∩𝒟, 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 {𝑋_𝑛,𝑛≥1} come from the common Pareto distribution with parameter 𝜅=1, 𝛽=2, 𝐹(𝑥;𝜅,𝛽)=1−𝜅𝜅+𝑥^𝛽,𝑥≥0,(4.1) which belongs to the class ℒ∩𝒟, and {(𝑋_2𝑛−1,𝑋_2𝑛),𝑛≥1} are independent replications of (𝑋₁,𝑋₂) with the joint distribution 𝐹_𝑋1,𝑋2(𝑥,𝑦)=−1𝛼ln1+𝑒^{−𝛼𝐹(𝑥)}−1𝑒^{−𝛼𝐹(𝑦)}−1𝑒^−𝛼−1,(4.2) with parameter 𝛼=1, where the joint distribution 𝐹_𝑋1,𝑋2(𝑥,𝑦) is constructed according to the Frank Copula. It has been proved in Example  4.2 of Liu [12] that 𝑋₁ and 𝑋₂ are END r.v.s. Since {(𝑋_2𝑛−1,𝑋_2𝑛),𝑛≥1} are independent copies of (𝑋₁,𝑋₂), the r.v.s {𝑋_𝑛,𝑛≥1} are END as well.

Assume that the claim arrival process 𝑁(𝑡) is the homogeneous Poisson process with intensity parameter 𝜆. Clearly, such an integer-valued process 𝑁(𝑡) satisfies the condition (2.8). Choose 𝜆=0.1. The total amount of premiums is simplified as 𝐶(𝑡)=𝑐𝑡 with the premium rate 𝑐=500, and the constant interest rate 𝛿=0.02. Here, we set the time 𝑇 as 𝑇=10 and the initial capital reserve 𝑥=500,10³,2×10³,5×10³, respectively. We aim to verify the accuracy of relation (2.4). The procedure of the computation of the finite-time ruin probability Ψ(𝑥,𝑇) in Theorem 2.2 is listed here.

Step 1. Assign a value for the variable 𝑥 and set 𝑙=0.

Step 2. Divide the close interval [0,𝑇] into 𝑚=1000 pieces, and denote each time point as 𝑡_𝑖, 𝑖=1,…,𝑚.

Step 3. For each 𝑡_𝑖, generate a random number 𝑛_𝑖 from the Poisson distribution 𝑃(𝜆𝑡_𝑖), and set 𝑛_𝑖 as the sample size of the claims.

Step 4. Generate the accident arrival time {𝜏^𝑖𝑘,𝑘=1,…,𝑛_𝑖} from the uniform distribution 𝑈(0,𝑡_𝑖) and the claim sizes {𝑋^𝑖𝑘,𝑘=1,…,𝑛_𝑖} from (4.1) and (4.2).

Step 5. Calculate the expression 𝐷 below for each 𝑡_𝑖 and denote them as {𝐷_𝑖,𝑖=1,…,𝑚}: 𝐷_𝑖=^𝑛𝑖_𝑘=1𝑋^𝑖𝑘𝑒^{−𝑟𝜏𝑖𝑘}−^𝑡𝑖₀𝑒^−𝑟𝑠𝐶(d𝑠),𝑖=1,…,𝑚,(4.3) where 𝑟 and 𝐶(𝑡) have been defined and their values have also been assigned.

Step 6. Select the maximum value from {𝐷_𝑖,𝑖=1,…,𝑚}, and denote it as 𝐷^∗, compare 𝐷^∗ with 𝑥; if 𝐷^∗>𝑥, then the value of 𝑙 increases 1.

Step 7. Repeat Step 2 through Step 6, 𝑁=10⁹ times.

Step 8. Calculate the moment estimate of the finite-time ruin probability, 𝑙/𝑁.

Step 9. Repeat Step 1 through Step 8 ten times and get ten estimates. Then, choose the median of the ten estimates as the analog value of the finite-time ruin probability.

For different value of 𝑥, 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 𝑥 becomes, the smaller the difference between the analog value and the theoretical result is. Therefore, the approximate relationship in Theorem 2.2 is reasonable.

Table 1 

Comparison between the analog value and the theoretical result in Theorem 2.2.

Acknowledgments

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.