Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 852852 |

Yang Yang, Xin Ma, Jin-guan Lin, "Approximation for the Finite-Time Ruin Probability of a General Risk Model with Constant Interest Rate and Extended Negatively Dependent Heavy-Tailed Claims", Mathematical Problems in Engineering, vol. 2011, Article ID 852852, 14 pages, 2011.

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

Academic Editor: P. Liatsis
Received06 Mar 2011
Revised03 May 2011
Accepted09 May 2011
Published14 Jul 2011


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 {š‘‹š‘›,š‘›ā‰„1} form a sequence of identically distributed, not necessarily independent, and nonnegative random variables (r.v.s) with common distribution š¹ such that š¹(š‘„)=1āˆ’š¹(š‘„)=P(š‘‹1>š‘„)>0 for all š‘„>0; the claim arrival process {š‘(š‘”),š‘”ā‰„0} is a general counting process, namely, a nonnegative, nondecreasing, right continuous, and integer-valued stochastic process with 0<Eš‘(š‘”)=šœ†(š‘”)<āˆž for all large š‘”>0. The times of the successive claims are denoted by {šœš‘›,š‘›ā‰„1}. The total amount of premiums accumulated up to time š‘”ā‰„0, denoted by š¶(š‘”) with š¶(0)=0 and š¶(š‘”)<āˆž almost surely for every š‘”>0, is another nonnegative and nondecreasing stochastic process. Assume that {š‘‹š‘›,š‘›ā‰„1}, {š‘(š‘”),š‘”ā‰„0} and {š¶(š‘”),š‘”ā‰„0} are mutually independent. Let š›æ>0 be the constant interest rate (i.e., after time š‘” one dollar becomes š‘’š›æš‘” dollars), and let š‘„ā‰„0 be the initial capital reserve of an insurance company. Then, the total discounted reserve up to time š‘”ā‰„0, denoted by š·(š‘”,š‘„), can be written as ī€œš·(š‘”,š‘„)=š‘„+š‘”0š‘’āˆ’š›æš‘ š¶(dš‘ )āˆ’š‘(š‘”)ī“š‘›=1š‘‹š‘›š‘’āˆ’š›æšœš‘›.(1.1) For a finite time š‘‡>0, the finite-time ruin probability is defined by īƒ©ĪØ(š‘„,š‘‡)=P(š·(š‘”,š‘„)<0,forsome0ā‰¤š‘”ā‰¤š‘‡)=Psupš‘”āˆˆ[0,š‘‡]īƒ©š‘(š‘”)ī“š‘›=1š‘‹š‘›š‘’āˆ’š›æšœš‘›āˆ’ī€œš‘”0š‘’āˆ’š›æš‘ īƒŖīƒŖ,š¶(dš‘ )>š‘„(1.2) while the ultimate ruin probability is defined by ĪØ(š‘„)=ĪØ(š‘„,āˆž)=P(š·(š‘”,š‘„)<0,forsomeš‘”ā‰„0).(1.3)

If the claim sizes {š‘‹š‘›,š‘›ā‰„1} are independent r.v.s, the model is called the independent general risk model, which was introduced by Wang [1]. In particular, if š¶(š‘”)=š‘š‘”, š‘”ā‰„0, with š‘>0 a deterministic constant and {š‘(š‘”),š‘”ā‰„0} is a Poisson process, then the model reduces to the classical one.

2. Introduction and Main Result

Hereafter, all limit relationships hold for š‘„ tending to āˆž unless otherwise stated. For two positive functions š‘“(š‘„) and š‘”(š‘„), we write š‘“(š‘„)āˆ¼š‘”(š‘„) if limš‘“(š‘„)/š‘”(š‘„)=1; write š‘“(š‘„)ā‰²š‘”(š‘„) if limsupš‘“(š‘„)/š‘”(š‘„)ā‰¤1 and š‘“(š‘„)=š‘œ(š‘”(š‘„)) if limš‘“(š‘„)/š‘”(š‘„)=0. The indicator function of an event š“ is denoted by šŸš“.

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 š¹ to be heavy tailed. A distribution š‘‰ is said to be dominatedly varying tailed, denoted by š‘‰āˆˆš’Ÿ, if limsupš‘‰(š‘„š‘¦)/š‘‰(š‘„)<āˆž for any š‘¦>0. A distribution š‘‰ is said to be long tailed, denoted by š‘‰āˆˆā„’, if limš‘‰(š‘„+š‘¦)/š‘‰(š‘„)=1 for any š‘¦>0. A distribution š‘‰ is said to be subexponential, denoted by š‘‰āˆˆš’®, if š‘‰š‘›āˆ—(š‘„)āˆ¼š‘›š‘‰(š‘„) for any š‘›ā‰„2, where š‘‰š‘›āˆ— denotes the š‘›-fold convolution of itself. A distribution š‘‰ is said to be regularly varying tailed, denoted by ā„›āˆ’š›¼,š›¼>0, if limš‘‰(š‘„š‘¦)/š‘‰(š‘„)=š‘¦āˆ’š›¼ for any š‘¦ā‰„1. A proper inclusion relationship holds that ā„›āˆ’š›¼āŠ‚ā„’āˆ©š’ŸāŠ‚š’®āŠ‚ā„’,(2.1) see, for example, Cline [2] or Embrechts and Omey [3]. For a distribution š‘‰, denote the upper Matuszewska index of the distribution š‘‰ by š½+š‘‰=āˆ’limš‘¦ā†’āˆžlogš‘‰āˆ—(š‘¦)logš‘¦withš‘‰āˆ—(š‘¦)=liminfš‘„ā†’āˆžš‘‰(š‘„š‘¦)š‘‰(š‘„),š‘¦>1.(2.2) In the terminology of Bingham et al. [4], the quantity š½+š‘‰ is actually the upper Matuszewska index of the function 1/š‘‰(š‘„), š‘„ā‰„0, as also pointed out in Tang and Tsitsiashvili [5]. Additionally, denote šæš‘‰=limš‘¦ā†˜1š‘‰āˆ—(š‘¦) (clearly, 0ā‰¤šæš‘‰ā‰¤1 ). The presented definitions yield that the following assertions are equivalent: (i)š‘‰āˆˆš’Ÿ,(ii)š‘‰āˆ—(š‘¦)>0forsomeš‘¦>1,(iii)šæš‘‰>0,(iv)š½+š‘‰<āˆž.(2.3)

The asymptotic behavior of the ruin probability in the classical risk model has been extensively investigated in the literature. KlĆ¼ppelberg and StadtmĆ¼ller [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 Corollaryā€‰ā€‰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 š¹āˆˆš’®, he obtained that for every š‘‡>0 for which šœ†(š‘‡)>0, ī€œĪØ(š‘„,š‘‡)āˆ¼š‘‡0āˆ’š¹ī€·š‘„š‘’š›æš‘”ī€øšœ†(dš‘”).(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 š¹āˆˆš’®; another is the general risk model, in which (2.4) also holds if š¹āˆˆā„’āˆ©š’Ÿ. We are interested in the latter, for example, the general risk model, and restate Theoremā€‰ā€‰ 2.2 of Wang [1] here.

Theorem 2.1. In the independent general risk model introduced in Section 1, assume that the claim sizes {š‘‹š‘›,š‘›ā‰„1} are independent and identically distributed nonnegative r.v.s with common distribution š¹āˆˆā„’āˆ©š’Ÿ. Assume that for any š‘‡>0 with šœ†(š‘‡)āˆ’šœ†(0)>0, there exists some constant šœ‚=šœ‚(š‘‡)>0 such that E(1+šœ‚)š‘(š‘‡)<āˆž.(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 š¹āˆˆš’Ÿ. Simultaneously, the condition (2.5) can be weakened to (2.8) below.

We call r.v.s {šœ‰š‘›,š‘›ā‰„1} are extended negatively dependent (END) if there exists some positive constant š‘€ such that both Pīƒ©š‘›ī™š‘˜=1ī€½šœ‰š‘˜>š‘¦š‘˜ī€¾īƒŖā‰¤š‘€š‘›ī‘š‘˜=1Pī€·šœ‰š‘˜>š‘¦š‘˜ī€øPīƒ©,(2.6)š‘›ī™š‘˜=1ī€½šœ‰š‘˜ā‰¤š‘¦š‘˜ī€¾īƒŖā‰¤š‘€š‘›ī‘š‘˜=1Pī€·šœ‰š‘˜ā‰¤š‘¦š‘˜ī€ø(2.7) hold for each š‘›ā‰„1 and all š‘¦1,ā€¦,š‘¦š‘›. This dependence structure was introduced by Liu [12]. Recall that r.v.s {šœ‰š‘›,š‘›ā‰„1} are called upper negatively dependent (UND) if (2.6) holds with š‘€=1, they are called lower negatively dependent (LND) if (2.7) holds with š‘€=1, and they are called negatively dependent (ND) if both (2.6) and (2.7) hold with š‘€=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 Exampleā€‰ā€‰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 {š‘‹š‘›,š‘›ā‰„1} are END nonnegative r.v.s with common distribution š¹āˆˆš’Ÿ and finite mean šœ‡. Assume that for any š‘‡>0 with šœ†(š‘‡)āˆ’šœ†(0)>0, there exists some constant š‘>š½+š¹ such that E(š‘(š‘‡))š‘<āˆž.(2.8) Then, it holds that šæš¹ī€œš‘‡0āˆ’š¹ī€·š‘„š‘’š›æš‘”ī€øšœ†(dš‘”)ā‰²ĪØ(š‘„,š‘‡)ā‰²šæš¹āˆ’1ī€œš‘‡0āˆ’š¹ī€·š‘„š‘’š›æš‘”ī€øšœ†(dš‘”).(2.9) Furthermore, if š¹āˆˆā„’āˆ©š’Ÿ, 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, š‘€ and š‘Ž 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 š’Ÿ and the END structure. The first lemma is a combination of Propositionā€‰ā€‰2.2.1 of Bingham et al. [4] and Lemmaā€‰ā€‰3.5 of Tang and Tsitsiashvili [15].

Lemma 3.1. If a distribution š‘‰āˆˆš’Ÿ, then (i)for any š›¾>š½+š‘‰, there exist positive constants š‘Ž and š‘ such that š‘‰(š‘¦)/š‘‰(š‘„)ā‰¤š‘Ž(š‘¦/š‘„)āˆ’š›¾ holds for all š‘„ā‰„š‘¦ā‰„š‘ and(ii) it holds for every š›¾>š½+š‘‰ that š‘„āˆ’š›¾=š‘œ(š‘‰(š‘„)).

By direct verification, END r.v.s have the following properties similar to those of ND r.v.s; see Lemmaā€‰ā€‰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 Lemmaā€‰ā€‰2.2 of Chen et al. [16].

Lemma 3.2. (i) If r.v.s {šœ‰š‘›,š‘›ā‰„1} are nonnegative and END, then for any š‘›ā‰„1, there exists some positive constant š‘€ such that āˆE(š‘›š‘˜=1šœ‰š‘˜āˆ)ā‰¤š‘€š‘›š‘˜=1Ešœ‰š‘˜.
(ii) If r.v.s {šœ‰š‘›,š‘›ā‰„1} are END and {š‘“š‘›(ā‹…),š‘›ā‰„1} are either all monotone increasing or all monotone decreasing, then {š‘“š‘›(šœ‰š‘›),š‘›ā‰„1} are still END.

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

Lemma 3.3. Let {šœ‰š‘›,š‘›ā‰„1} be identically distributed and END r.v.s with common distribution š‘‰ and šœ‡+š‘‰=Ešœ‰1šŸ{šœ‰1ā‰„0}<āˆž. Then, for any šœƒ>0, š‘„>0 and š‘›ā‰„1, there exists some positive constant š‘€ such that Pīƒ©š‘›ī“š‘˜=1šœ‰š‘˜īƒŖ>š‘„ā‰¤š‘›š‘‰īƒ©(šœƒš‘„)+š‘€š‘’šœ‡+š‘‰š‘›š‘„īƒŖšœƒāˆ’1.(3.1)

Proof. Following the proof of Lemmaā€‰ā€‰2.3 of Tang [17], we employ a standard truncation argument to prove this lemma. For simplicity, we write š‘†šœ‰š‘›=āˆ‘š‘›š‘˜=1šœ‰š‘˜, š‘›ā‰„1. If šœ‡+š‘‰=0, then šœ‰š‘› is almost surely nonpositive for each š‘›ā‰„1, implying P(š‘†šœ‰š‘›>š‘„)=0 for any positive š‘„, and thus (3.1) holds.
Let, in the following, šœ‡+š‘‰>0. For any fixed šœƒ>0 and positive integer š‘›, define Ģƒšœ‰š‘›ī€½šœ‰=minš‘›ī€¾,Ģƒšœ‰,šœƒš‘„+š‘›ī€½Ģƒšœ‰=maxš‘›ī€¾,0=šœ‰š‘›šŸ{0ā‰¤šœ‰š‘›ā‰¤šœƒš‘„}+šœƒš‘„šŸ{šœ‰š‘›>šœƒš‘„}.(3.2) According to Lemma 3.2(ii), {Ģƒšœ‰š‘›,š‘›ā‰„1} and {Ģƒšœ‰+š‘›,š‘›ā‰„1} are still END r.v.s, respectively. Denote ī‚š‘†šœ‰š‘›=āˆ‘š‘›š‘˜=1Ģƒšœ‰š‘˜, š‘›ā‰„1. Clearly, Pī‚€š‘†šœ‰š‘›ī‚ī‚µš‘†>š‘„=Pšœ‰š‘›>š‘„,max1ā‰¤š‘˜ā‰¤š‘›šœ‰š‘˜ī‚¶ī‚µš‘†>šœƒš‘„+Pšœ‰š‘›>š‘„,max1ā‰¤š‘˜ā‰¤š‘›šœ‰š‘˜ī‚¶ā‰¤šœƒš‘„ā‰¤š‘›ī‚€ī‚š‘†š‘‰(šœƒš‘„)+Pšœ‰š‘›ī‚.>š‘„(3.3) It remains to estimate the second summand in (3.3). For a positive ā„Ž, by Lemma 3.2(ii), {š‘’ā„ŽĢƒšœ‰+š‘›,š‘›ā‰„1} are END nonnegative r.v.s. Hence, using identity Eš‘’ā„ŽĢƒšœ‰+1=ī€œ0šœƒš‘„ī€·š‘’ā„Žš‘¢ī€øī€·š‘’āˆ’1š‘‰(dš‘¢)+ā„Žšœƒš‘„ī€øāˆ’1š‘‰(šœƒš‘„)+1,(3.4) by Markov inequality and Lemma 3.2(i) we have Pī‚€ī‚š‘†šœ‰š‘›ī‚>š‘„ā‰¤š‘’āˆ’ā„Žš‘„Eš‘’ā„Žī‚š‘†šœ‰š‘›ā‰¤š‘’āˆ’ā„Žš‘„Eš‘’ā„Žāˆ‘š‘›š‘˜=1Ģƒšœ‰+š‘˜ā‰¤š‘’āˆ’ā„Žš‘„š‘€(Eš‘’ā„ŽĢƒšœ‰+1)š‘›=š‘€š‘’āˆ’ā„Žš‘„ī‚µī€œ0šœƒš‘„ī€·š‘’ā„Žš‘¢ī€øī€·š‘’āˆ’1š‘‰(dš‘¢)+ā„Žšœƒš‘„ī€øāˆ’1ī‚¶š‘‰(šœƒš‘„)+1š‘›.(3.5) Since 1+š‘¢ā‰¤š‘’š‘¢ for all š‘¢āˆˆā„ and (š‘’ā„Žš‘¢āˆ’1)/š‘¢ is strictly increasing in š‘¢>0, from (3.5), we obtain Pī‚€ī‚š‘†šœ‰š‘›ī‚ī‚»š‘›ī€œ>š‘„ā‰¤š‘€exp0šœƒš‘„š‘’ā„Žš‘¢āˆ’1š‘¢ī€·š‘’š‘¢š‘‰(dš‘¢)+š‘›ā„Žšœƒš‘„ī€øāˆ’1ī‚¼ī‚»š‘›š‘’š‘‰(šœƒš‘„)āˆ’ā„Žš‘„ā‰¤š‘€expā„Žšœƒš‘„āˆ’1ī‚µī€œšœƒš‘„0šœƒš‘„š‘¢š‘‰(dš‘¢)+šœƒš‘„ī‚¶ī‚¼ī‚»š‘›š‘’š‘‰(šœƒš‘„)āˆ’ā„Žš‘„ā‰¤š‘€expā„Žšœƒš‘„āˆ’1šœ‡šœƒš‘„+š‘‰ī‚¼.āˆ’ā„Žš‘„(3.6) Choose ā„Ž=(šœƒš‘„)āˆ’1log(š‘„(šœ‡+š‘‰š‘›)āˆ’1+1), which is positive. For such ā„Ž, by (3.6), we have Pī‚€ī‚š‘†šœ‰š‘›ī‚īƒÆ1>š‘„ā‰¤š‘€expšœƒāˆ’1šœƒīƒ©š‘„logšœ‡+š‘‰š‘›īƒÆ1+1īƒŖīƒ°ā‰¤š‘€expšœƒlogš‘’šœ‡+š‘‰š‘›š‘„īƒ°.(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 {š‘‹š‘›,š‘›ā‰„1} are END nonnegative r.v.s with common distribution š¹āˆˆš’Ÿ. Let š‘ be an arbitrary nonnegative r.v. and assume that {š‘‹š‘›,š‘›ā‰„1}, {š‘(š‘”),š‘”ā‰„0} and š‘ are mutually independent. Then, for any š‘‡>0 and any positive integer š‘›0, šæš¹š‘›0ī“š‘˜š‘˜=1ī“š‘—=1Pī€·š‘‹š‘—š‘’āˆ’š›æšœš‘—ī€øā‰²>š‘„,š‘(š‘‡)=š‘˜š‘›0ī“š‘˜=1Pīƒ©š‘˜ī“š‘—=1š‘‹š‘—š‘’āˆ’š›æšœš‘—īƒŖ>š‘„+š‘,š‘(š‘‡)=š‘˜ā‰²šæš¹š‘›āˆ’10ī“š‘˜š‘˜=1ī“š‘—=1Pī€·š‘‹š‘—š‘’āˆ’š›æšœš‘—ī€ø.>š‘„,š‘(š‘‡)=š‘˜(3.8) Furthermore, if š¹āˆˆā„’āˆ©š’Ÿ, then š‘›0ī“š‘˜=1Pīƒ©š‘˜ī“š‘—=1š‘‹š‘—š‘’āˆ’š›æšœš‘—īƒŖāˆ¼>š‘„+š‘,š‘(š‘‡)=š‘˜š‘›0ī“š‘˜š‘˜=1ī“š‘—=1Pī€·š‘‹š‘—š‘’āˆ’š›æšœš‘—ī€ø>š‘„,š‘(š‘‡)=š‘˜.(3.9)

We remark that if š¹ is consistently varying tailed (see the definition in Chen and Yuen [18]), then by conditioning (3.9) easily follows from Theoremā€‰ā€‰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 (šœ1,ā€¦,šœš‘›0) and š‘.

Proof. We follow the line of the proof of Lemmaā€‰ā€‰3.6 of Wang [1] with some modifications in relation to the properties of the class š’Ÿ and the END structure. Clearly, for each š‘˜=1,ā€¦,š‘›0, Pīƒ©š‘˜ī“š‘—=1š‘‹š‘—š‘’āˆ’š›æšœš‘—īƒŖ=ī€œ>š‘„+š‘,š‘(š‘‡)=š‘˜{0ā‰¤š‘”1ā‰¤ā‹Æā‰¤š‘”š‘˜ā‰¤š‘‡,š‘”š‘˜+1>š‘‡}ī€œāˆž0āˆ’Pīƒ©š‘˜ī“š‘—=1š‘‹š‘—š‘’āˆ’š›æš‘”š‘—īƒŖī€·>š‘„+š‘§Ć—Pš‘āˆˆdš‘§,šœ1āˆˆdš‘”1,ā€¦,šœš‘˜+1āˆˆdš‘”š‘˜+1ī€ø.(3.10) We first show the upper bound. For any fixed š‘™>0, Pīƒ©š‘˜ī“š‘—=1š‘‹š‘—š‘’āˆ’š›æš‘”š‘—īƒŖīƒ©>š‘„+š‘§ā‰¤Pš‘˜īšš‘—=1ī€½š‘‹š‘—š‘’āˆ’š›æš‘”š‘—ī€¾īƒŖīƒ©>š‘„+š‘§āˆ’š‘™+Pš‘˜ī“š‘—=1š‘‹š‘—š‘’āˆ’š›æš‘”š‘—>š‘„+š‘§,max1ā‰¤š‘—ā‰¤š‘˜š‘‹š‘—š‘’āˆ’š›æš‘”š‘—īƒŖā‰¤š‘„+š‘§āˆ’š‘™āˆ¶=š¼1+š¼2.(3.11) By š¹āˆˆš’Ÿ, for any 0<šœƒ<1 and each š‘˜=1,ā€¦,š‘›0, we have uniformly for all š‘”1,ā€¦,š‘”š‘˜āˆˆ[0,š‘‡] and š‘§āˆˆ[0,āˆž), š¼1ā‰¤š‘˜ī“š‘—=1š¹ī€·šœƒ(š‘„+š‘§)š‘’š›æš‘”š‘—ī€øā‰²šæš¹š‘˜āˆ’1ī“š‘—=1š¹ī€·(š‘„+š‘§)š‘’š›æš‘”š‘—ī€ø,(3.12) by firstly letting š‘„ā†’āˆž then šœƒā†—1. We note that {š‘‹š‘›,š‘›ā‰„1} are END r.v.s. Then, by š¹āˆˆš’Ÿ, there exists some positive constant š‘€=š‘€(š‘›0) such that for sufficiently large š‘„, each š‘˜=1,ā€¦,š‘›0, all š‘”1,ā€¦,š‘”š‘˜āˆˆ[0,š‘‡] and š‘§āˆˆ[0,āˆž), š¼2īƒ©=Pš‘˜ī“š‘—=1š‘‹š‘—š‘’āˆ’š›æš‘”š‘—>š‘„+š‘§,š‘„+š‘§š‘˜<max1ā‰¤š‘—ā‰¤š‘˜š‘‹š‘—š‘’āˆ’š›æš‘”š‘—īƒŖīƒ©ā‰¤š‘„+š‘§āˆ’š‘™ā‰¤Pš‘˜īšš‘–=1īƒÆī“š‘—ā‰ š‘–š‘‹š‘—š‘’āˆ’š›æš‘”š‘—>š‘™,š‘‹š‘–š‘’āˆ’š›æš‘”š‘–>š‘„+š‘§š‘˜ā‰¤īƒ°īƒŖš‘˜ī“š‘–=1ī“š‘—ā‰ š‘–Pī‚€š‘‹š‘—š‘’āˆ’š›æš‘”š‘—>š‘™š‘˜āˆ’1,š‘‹š‘–š‘’āˆ’š›æš‘”š‘–>š‘„+š‘§š‘˜ī‚ā‰¤š‘€š‘˜ī“š‘–=1ī“š‘—ā‰ š‘–š¹ī‚µš‘™š‘’š›æš‘”š‘—ī‚¶š‘˜āˆ’1š¹ī‚µ(š‘„+š‘§)š‘’š›æš‘”š‘–š‘˜ī‚¶ā‰¤š‘€š¹ī‚µš‘™š‘›0ī‚¶āˆ’1š‘˜ī“š‘—=1š¹ī€·(š‘„+š‘§)š‘’š›æš‘”š‘—ī€ø.(3.13) Since š‘™ can be arbitrarily large, it follows that limsupš‘™ā†’āˆžlimsupš‘„ā†’āˆžsupš‘”1,ā€¦,š‘”š‘˜āˆˆ[][0,š‘‡,š‘§āˆˆ0,āˆž)š¼2āˆ‘š‘˜š‘—=1š¹ī€·(š‘„+š‘§)š‘’š›æš‘”š‘—ī€ø=0.(3.14) Hence, from (3.10)ā€“(3.14), we obtain for each š‘˜=1,ā€¦,š‘›0, Pīƒ©š‘˜ī“š‘—=1š‘‹š‘—š‘’āˆ’š›æšœš‘—īƒŖ>š‘„+š‘,š‘(š‘‡)=š‘˜ā‰²šæš¹š‘˜āˆ’1ī“š‘—=1ī€œ{0ā‰¤š‘”1ā‰¤ā‹Æā‰¤š‘”š‘˜ā‰¤š‘‡,š‘”š‘˜+1>š‘‡}ī€œāˆž0āˆ’š¹ī€·(š‘„+š‘§)š‘’š›æš‘”š‘—ī€øī€·Ć—Pš‘āˆˆdš‘§,šœ1āˆˆdš‘”1,ā€¦,šœš‘˜+1āˆˆdš‘”š‘˜+1ī€ø=šæš¹š‘˜āˆ’1ī“š‘—=1Pī€·š‘‹š‘—š‘’āˆ’š›æšœš‘—ī€ø>š‘„+š‘,š‘(š‘‡)=š‘˜ā‰¤šæš¹š‘˜āˆ’1ī“š‘—=1Pī€·š‘‹š‘—š‘’āˆ’š›æšœš‘—ī€ø.>š‘„,š‘(š‘‡)=š‘˜(3.15) As for the lower bound for (3.10), since {š‘‹š‘›,š‘›ā‰„1} are END r.v.s, we have for sufficiently large š‘„ and each š‘˜=1,ā€¦,š‘›0, Pīƒ©š‘˜ī“š‘—=1š‘‹š‘—š‘’āˆ’š›æš‘”š‘—īƒŖīƒ©>š‘„+š‘§ā‰„Pš‘˜īšš‘—=1ī€½š‘‹š‘—š‘’āˆ’š›æš‘”š‘—ī€¾īƒŖā‰„>š‘„+š‘§š‘˜ī“š‘—=1š¹ī€·(š‘„+š‘§)š‘’š›æš‘”š‘—ī€øāˆ’ī“1ā‰¤š‘–<š‘—ā‰¤š‘˜Pī€·š‘‹š‘–š‘’āˆ’š›æš‘”š‘–>š‘„+š‘§,š‘‹š‘—š‘’āˆ’š›æš‘”š‘—ī€øā‰„>š‘„+š‘§š‘˜ī“š‘—=1š¹ī€·(š‘„+š‘§)š‘’š›æš‘”š‘—ī€øī“āˆ’š‘€1ā‰¤š‘–<š‘—ā‰¤š‘˜š¹ī€·(š‘„+š‘§)š‘’š›æš‘”š‘–ī€øš¹ī€·(š‘„+š‘§)š‘’š›æš‘”š‘—ī€ø=(1āˆ’š‘œ(1))š‘˜ī“š‘—=1š¹ī€·(š‘„+š‘§)š‘’š‘Ÿš‘”š‘—ī€ø(3.16) holds uniformly for all š‘”1,ā€¦,š‘”š‘˜āˆˆ[0,š‘‡] and š‘§āˆˆ[0,āˆž). By š¹āˆˆš’Ÿ and Fatou's lemma, we have for any Ģƒšœƒ>1 and all š‘—=1,2,ā€¦, 1liminfPī€·š‘‹š¹(š‘„)š‘—>š‘„+š‘š‘’š›æš‘‡ī€øī€œ=liminfāˆž0āˆ’š¹ī€·š‘„+š‘§š‘’š›æš‘‡ī€øā‰„ī€œš¹(š‘„)P(š‘āˆˆdš‘§)āˆž0āˆ’liminfš¹ī€·Ģƒī€øšœƒš‘„š¹=(š‘„)P(š‘āˆˆdš‘§)š¹āˆ—ī€·Ģƒšœƒī€øāŸ¶šæš¹,Ģƒšœƒā†˜1,(3.17) which means Pī€·š‘‹š‘—>š‘„+š‘š‘’š›æš‘‡ī€øā‰³šæš¹š¹(š‘„).(3.18) Similar to (3.15), from (3.10), (3.16), and (3.18), we obtain for each š‘˜=1,ā€¦,š‘›0, Pīƒ©š‘˜ī“š‘—=1š‘‹š‘—š‘’āˆ’š›æšœš‘—īƒŖā‰³>š‘„+š‘,š‘(š‘‡)=š‘˜š‘˜ī“š‘—=1Pī€·š‘‹š‘—š‘’āˆ’š›æšœš‘—ī€øā‰„>š‘„+š‘,š‘(š‘‡)=š‘˜š‘˜ī“š‘—=1ī€œ{0ā‰¤š‘”1ā‰¤ā€¦ā‰¤š‘”š‘˜ā‰¤š‘‡,š‘”š‘˜+1>š‘‡}Pī€·š‘‹š‘—>š‘„š‘’š›æš‘”š‘—+š‘š‘’š›æš‘‡ī€øPī€·šœ1āˆˆdš‘”1,ā€¦,šœš‘˜+1āˆˆdš‘”š‘˜+1ī€øā‰³šæš¹š‘˜ī“š‘—=1ī€œ{0ā‰¤š‘”1ā‰¤ā€¦ā‰¤š‘”š‘˜ā‰¤š‘‡,š‘”š‘˜+1>š‘‡}š¹ī€·š‘„š‘’š›æš‘”š‘—ī€øPī€·šœ1āˆˆdš‘”1,ā€¦,šœš‘˜+1āˆˆdš‘”š‘˜+1ī€ø=šæš¹š‘˜ī“š‘—=1Pī€·š‘‹š‘—š‘’āˆ’š›æšœš‘—ī€ø.>š‘„,š‘(š‘‡)=š‘˜(3.19) The desired relation (3.8) follows now from (3.15) and (3.19).
If š¹āˆˆā„’āˆ©š’Ÿ, (3.9) follows by using the properties of the class ā„’ 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 Theoremā€‰ā€‰2.2 of Wang [1] (e.g., Theorem 2.1 of this paper) to prove this result. Clearly, š¹āˆˆš’Ÿ and šœ‡<āˆž imply š½+š¹ā‰„1. By (2.8), we have for any šœ–>0, there exists some positive integer š‘›1=š‘›1(š‘‡,šœ–) such that E(š‘(š‘‡))š‘šŸ{š‘(š‘‡)>š‘›1}ā‰¤šœ–.(3.20)
To estimate the upper bound of ĪØ(š‘„,š‘‡), we split it into two parts as īƒ©ĪØ(š‘„,š‘‡)ā‰¤Pš‘(š‘‡)ī“š‘—=1š‘‹š‘—š‘’āˆ’š›æšœš‘—īƒŖ=āŽ›āŽœāŽœāŽ>š‘„š‘›1ī“š‘˜=1+āˆžī“š‘˜=š‘›1+1āŽžāŽŸāŽŸāŽ Pīƒ©š‘˜ī“š‘—=1š‘‹š‘—š‘’āˆ’š›æšœš‘—īƒŖ>š‘„,š‘(š‘‡)=š‘˜āˆ¶=š¼3+š¼4.(3.21) According to Lemma 3.4 of this paper and Lemmaā€‰ā€‰3.5 of Wang [1], we have for sufficiently large š‘„, š¼3ā‰¤(1+šœ–)šæš¹š‘›āˆ’11ī“š‘˜š‘˜=1ī“š‘—=1Pī€·š‘‹š‘—š‘’āˆ’š›æšœš‘—ī€ø>š‘„,š‘(š‘‡)=š‘˜ā‰¤(1+šœ–)šæš¹āˆžāˆ’1ī“š‘—=1Pī€·š‘‹š‘—š‘’āˆ’š›æšœš‘—ī€ø>š‘„,š‘(š‘‡)ā‰„š‘—=(1+šœ–)šæš¹āˆžāˆ’1ī“š‘—=1Pī€·š‘‹š‘—š‘’āˆ’š›æšœš‘—>š‘„,šœš‘—ī€ø=ā‰¤š‘‡(1+šœ–)šæš¹āˆ’1ī€œš‘‡0āˆ’š¹ī€·š‘„š‘’š›æš‘”ī€øšœ†(dš‘”).(3.22) By Lemma 3.3, š¹āˆˆš’Ÿ, Lemma 3.1(ii), (3.20), and š‘>š½+š¹ā‰„1, there exists some positive constant š‘€ such that for sufficiently large š‘„, š¼4ā‰¤āˆžī“š‘˜=š‘›1+1Pīƒ©š‘˜ī“š‘—=1š‘‹š‘—īƒŖā‰¤>š‘„P(š‘(š‘‡)=š‘˜)š¹ī€·š‘āˆ’1š‘„ī€øāˆžī“š‘˜=š‘›1+1š‘˜P(š‘(š‘‡)=š‘˜)+š‘€(š‘’šœ‡)š‘š‘„āˆžāˆ’š‘ī“š‘˜=š‘›1+1š‘˜š‘P(š‘(š‘‡)=š‘˜)ā‰¤š‘€š¹ī€·(š‘„)Eš‘(š‘‡)šŸ{š‘(š‘‡)>š‘›1}+E(š‘(š‘”))š‘šŸ{š‘(š‘‡)>š‘›1}ī€ø=š‘€šœ–š¹(š‘„).(3.23) By Lemma 3.1(i), for any š›¾>š½+š¹, there exists some positive constant š‘Ž such that for sufficiently large š‘„, ī€œš‘‡0āˆ’š¹ī€·š‘„š‘’š›æš‘”ī€øšœ†(dš‘”)ā‰„š‘Žāˆ’1ī€œš¹(š‘„)š‘‡0āˆ’š‘’āˆ’š›¾š›æš‘”šœ†(dš‘”)ā‰„š‘Žāˆ’1š‘’āˆ’š›¾š›æš‘‡(šœ†(š‘‡)āˆ’šœ†(0))š¹(š‘„),(3.24) which, combining (3.23) and šœ†(š‘‡)āˆ’šœ†(0)>0, implies š¼4ī€œā‰¤š‘€šœ–š‘‡0āˆ’š¹ī€·š‘„š‘’š›æš‘”ī€øšœ†(dš‘”).(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 ĪØ(š‘„,š‘‡), by Lemma 3.4, we have for the above given šœ–>0 and sufficiently large š‘„, īƒ©ĪØ(š‘„,š‘‡)ā‰„Pš‘(š‘‡)ī“š‘—=1š‘‹š‘—š‘’āˆ’š›æšœš‘—ī€œ>š‘„+š‘‡0š‘’āˆ’š›æš‘ īƒŖā‰„š¶(dš‘ )š‘›1ī“š‘˜=1Pīƒ©š‘˜ī“š‘—=1š‘‹š‘—š‘’āˆ’š›æšœš‘—ī€œ>š‘„+š‘‡0š‘’āˆ’š›æš‘ īƒŖš¶(dš‘ ),š‘(š‘‡)=š‘˜ā‰„(1āˆ’šœ–)šæš¹š‘›1ī“š‘˜š‘˜=1ī“š‘—=1Pī€·š‘‹š‘—š‘’āˆ’š›æšœš‘—ī€ø>š‘„,š‘(š‘‡)=š‘˜=(1āˆ’šœ–)šæš¹āŽ›āŽœāŽœāŽāˆžī“š‘—=1Pī€·š‘‹š‘—š‘’āˆ’š›æšœš‘—>š‘„,šœš‘—ī€øāˆ’ā‰¤š‘‡āˆžī“š‘˜=š‘›1š‘˜+1ī“š‘—=1Pī€·š‘‹š‘—š‘’āˆ’š›æšœš‘—ī€øāŽžāŽŸāŽŸāŽ >š‘„,š‘(š‘‡)=š‘˜āˆ¶=(1āˆ’šœ–)šæš¹ī‚µī€œš‘‡0āˆ’š¹ī€·š‘„š‘’š›æš‘”ī€øšœ†(dš‘”)āˆ’š¼5ī‚¶.(3.26) Analogously to the estimate for š¼4, we have for sufficiently large š‘„, š¼5ā‰¤š¹(š‘„)Eš‘(š‘‡)šŸ{š‘(š‘‡)>š‘›1}ī€œā‰¤š‘€šœ–š‘‡0āˆ’š¹ī€·š‘„š‘’š›æš‘”ī€øšœ†(dš‘”).(3.27) From (3.26) and (3.27), we obtain the left-hand side of (2.9).
If š¹āˆˆā„’āˆ©š’Ÿ, 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 {š‘‹š‘›,š‘›ā‰„1} come from the common Pareto distribution with parameter šœ…=1, š›½=2, ī‚€šœ…š¹(š‘„;šœ…,š›½)=1āˆ’ī‚šœ…+š‘„š›½,š‘„ā‰„0,(4.1) which belongs to the class ā„’āˆ©š’Ÿ, and {(š‘‹2š‘›āˆ’1,š‘‹2š‘›),š‘›ā‰„1} are independent replications of (š‘‹1,š‘‹2) with the joint distribution š¹š‘‹1,š‘‹21(š‘„,š‘¦)=āˆ’š›¼īƒ©ī€·š‘’ln1+āˆ’š›¼š¹(š‘„)š‘’āˆ’1ī€øī€·āˆ’š›¼š¹(š‘¦)ī€øāˆ’1š‘’āˆ’š›¼īƒŖāˆ’1,(4.2) with parameter š›¼=1, where the joint distribution š¹š‘‹1,š‘‹2(š‘„,š‘¦) is constructed according to the Frank Copula. It has been proved in Exampleā€‰ā€‰4.2 of Liu [12] that š‘‹1 and š‘‹2 are END r.v.s. Since {(š‘‹2š‘›āˆ’1,š‘‹2š‘›),š‘›ā‰„1} are independent copies of (š‘‹1,š‘‹2), the r.v.s {š‘‹š‘›,š‘›ā‰„1} are END as well.

Assume that the claim arrival process š‘(š‘”) is the homogeneous Poisson process with intensity parameter šœ†. Clearly, such an integer-valued process š‘(š‘”) satisfies the condition (2.8). Choose šœ†=0.1. The total amount of premiums is simplified as š¶(š‘”)=š‘š‘” with the premium rate š‘=500, and the constant interest rate š›æ=0.02. Here, we set the time š‘‡ as š‘‡=10 and the initial capital reserve š‘„=500,103,2Ɨ103,5Ɨ103, respectively. We aim to verify the accuracy of relation (2.4). The procedure of the computation of the finite-time ruin probability ĪØ(š‘„,š‘‡) in Theorem 2.2 is listed here.

Step 1. Assign a value for the variable š‘„ and set š‘™=0.

Step 2. Divide the close interval [0,š‘‡] into š‘š=1000 pieces, and denote each time point as š‘”š‘–, š‘–=1,ā€¦,š‘š.

Step 3. For each š‘”š‘–, generate a random number š‘›š‘– from the Poisson distribution š‘ƒ(šœ†š‘”š‘–), and set š‘›š‘– as the sample size of the claims.

Step 4. Generate the accident arrival time {šœš‘–š‘˜,š‘˜=1,ā€¦,š‘›š‘–} from the uniform distribution š‘ˆ(0,š‘”š‘–) and the claim sizes {š‘‹š‘–š‘˜,š‘˜=1,ā€¦,š‘›š‘–} from (4.1) and (4.2).

Step 5. Calculate the expression š· below for each š‘”š‘– and denote them as {š·š‘–,š‘–=1,ā€¦,š‘š}: š·š‘–=š‘›š‘–ī“š‘˜=1š‘‹š‘–š‘˜š‘’āˆ’š‘Ÿšœš‘–š‘˜āˆ’ī€œš‘”š‘–0š‘’āˆ’š‘Ÿš‘ š¶(dš‘ ),š‘–=1,ā€¦,š‘š,(4.3) where š‘Ÿ and š¶(š‘”) have been defined and their values have also been assigned.

Step 6. Select the maximum value from {š·š‘–,š‘–=1,ā€¦,š‘š}, and denote it as š·āˆ—, compare š·āˆ— with š‘„; if š·āˆ—>š‘„, then the value of š‘™ increases 1.

Step 7. Repeat Step 2 through Step 6, š‘=109 times.

Step 8. Calculate the moment estimate of the finite-time ruin probability, š‘™/š‘.

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 š‘„, 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 š‘„ becomes, the smaller the difference between the analog value and the theoretical result is. Therefore, the approximate relationship in Theorem 2.2 is reasonable.

š‘„ ( Ɨ 1 0 3 ) Theoretical result Analog value

0.5 3 . 2 8 4 6 š‘’ āˆ’ 6 3 . 8 1 2 0 š‘’ āˆ’ 6 ( 1 6 . 1 % )
1 8 . 2 2 7 0 š‘’ āˆ’ 7 9 . 1 1 0 0 š‘’ āˆ’ 7 ( 1 0 . 7 % )
2 2 . 0 5 8 6 š‘’ āˆ’ 7 2 . 2 3 0 0 š‘’ āˆ’ 7 ( 8 . 3 % )
5 3 . 2 9 5 6 š‘’ āˆ’ 8 3 . 5 0 0 0 š‘’ āˆ’ 8 ( 6 . 2 % )


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. ā€‰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.


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