ASYMPTOTIC BEHAVIOR OF THE FINITE-TIME RUIN PROBABILITY WITH PAIRWISE QUASI-ASYMPTOTICALLY INDEPENDENT CLAIMS AND CONSTANT INTEREST FORCE

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1 ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 44, Number 5, 214 ASYMPTOTIC BEHAVIOR OF THE FINITE-TIME RUIN PROBABILITY WITH PAIRWISE QUASI-ASYMPTOTICALLY INDEPENDENT CLAIMS AND CONSTANT INTEREST FORCE QINGWU GAO, NA JIN AND HOUCAI SHEN ABSTRACT. This paper will obtain an asymptotic formula of the finite-time ruin probability in a generalized risk model with constant interest force, in which the claim sizes are pairwise quasi-asymptotically independent and their arrival process is an arbitrary counting process. In particular, when the claim inter-arrival times follow a certain dependence structure, the result obtained can also include an asymptotic formula for the infinite-time ruin probability. 1. Risk model. In this paper, we investigate the finite-time ruin probability in a generalized risk model with constant interest force, where the claim sizes {X i, i 1} form a sequence of nonnegative, identically distributed, but not necessarily independent, random variables r.v.s with common distribution F, and the claim arrival process {Nt, t } is a general counting process, independent of {X i, i 1}. Hence, the aggregate claim amount up to time t is expressed as St = Nt X i Keywords and phrases. Asymptotics, finite-time ruin probability, pairwise quasiasymptotic independence, consistent variation. The work of the first author was supported by the National Natural Science Foundation of China Grant Nos , and , the Natural Science Foundation of Jiangsu Higher Education Institutions of China Grant Nos. 13KJB1114 and 14JB1114, the Projects Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions and the Key Lab of Financial Engineering of Jiangsu Province. The work of the third author was partly supported by the National Natural Science Foundation of China Grant No and MOE Ministry of Education in China Project Grant No The third author is the corresponding author. Received by the editors on February 12, 212, and in revised form on July 7, 212. DOI:1.1216/RMJ Copyright c 214 Rocky Mountain Mathematics Consortium 155

2 156 QINGWU GAO, NA JIN AND HOUCAI SHEN with St = if Nt =. Assume that the total amount of premiums accumulated before time t, denoted by Ct, is a nonnegative and nondecreasing stochastic process with C = and Ct < almost surely a.s. for every t <. Let r < be the constant interest force and x < the insurer s initial reserve. Thus, the total reserve up to time t < of the insurance company, denoted by U r t, satisfies 1.1 U r t = xe rt + t e rt s Cds t e rt s Sds. Apparently, by the conditions on Ct, one can easily know that, for any fixed < t <, 1.2 Ct = t e rs Cds < almost surely, where Ct is the discounted value of premiums accumulated before time t. The ruin probability within a finite time T is defined by 1.3 ψ r x, T = P U r t < for some t T, and the infinite-time ruin probability is 1.4 ψ r x, = P U r t < for some t <. For later use, let {θ i, i 1} denote the claim inter-arrival times, and then let τ k = k θ i, k 1, denote the arrival times of successive claims, which can constitute a counting process 1.5 Nt = 1 {τk t}, t, k=1 where 1 A is the indicator function of an event A. If {θ i, i 1} are independent and identically distributed i.i.d. r.v.s, then {Nt, t } is the renewal process and the risk model with {Nt, t } a renewal process which was studied by Tang [18], Chen and Ng [2], Hao and Tang [7] and some others. If {θ i, i 1} follows a certain dependence structure, then {Nt, t } is a quasi-renewal process and the risk model with {Nt, t } a quasi-renewal process was studied by Li et al. [13], Yang and Wang [25], Wang et al. [23], Liu et al. [14] and so on. However, for the model of this paper {Nt, t } is an arbitrary

3 FINITE-TIME RUIN PROBABILITY 157 counting process, and thus no assumption is made on the dependence structure of {θ i, i 1}, also see Wang [22]. 2. Introduction and main results. Throughout this paper, all limit relationships are taken as x unless stated otherwise. For two positive functions a and b, we write ax = Obx if lim sup ax/bx = C <, write ax bx or bx ax if C 1, write ax bx if ax bx and bx ax, and write ax = obx if C =. As generally admitted, in the insurance industry how to model dangerous claim sizes is one of the main worries of practicing actuaries, and actually most practitioners choose the claim-size distribution from the heavy-tailed distribution class, one of which is the subexponential class. Say that a distribution V belongs to the subexponential class, denoted by V S, if V x = 1 V x > for all x > and V 2 x 2V x, where V 2 denotes the 2-fold convolution of V. Clearly, if V S then V is long-tailed, denoted by V L and characterized by V x + y V x for all y. Another important class of heavy-tailed distributions is the dominated variation class, say that a distribution V belongs to the dominated variation class, denoted by V D, if V xy = OV x for all y >. A slightly smaller class of D is the consistent variation class, say that a distribution V belongs to the consistent variation class, denoted by V C, where or equivalently, lim y 1 lim y 1 lim inf V xy/v x = 1, lim sup V xy/v x = 1. It is well known that the following inclusion relationships are valid and proper: C L D S L.

4 158 QINGWU GAO, NA JIN AND HOUCAI SHEN For a distribution V and y >, we set J + V = lim log V y/ log y with V y = lim inf V xy/v x y and J V = lim log V y/ log y with V y = lim sup V xy/v x. y For more details of heavy-tailed distributions and their applications to insurance and finance, the readers are referred to Bingham et al. [1] and Embrechts et al. [4]. For the risk model with constant interest force and i.i.dḣeavy-tailed claims, there are many results on the asymptotic behavior for the finitetime ruin probability ψ r x, T, < T, see Klüppelberg and Stadtmüller [1], Kalashnikov and Konstantinides [9], Konstantinides et al. [12], Tang [18, 19], Wang [22], Hao and Tang [7], among others. In particular, Wang [22] showed that, in a risk model, the claim sizes {X i, i 1} are i.i.d. and nonnegative r.v.s with common distribution F L D, and their arrival process {Nt, t } is a general counting process satisfying ENT > and E1 + δ NT < for any fixed < T < and some δ = δt >, and {X i, i 1}, {Nt, t } and {Ct, t } are mutually independent. Then, for fixed < T <, 2.1 ψ r x, T F xe rt dent. But, in most practical situations the independence assumption on claim sizes is unrealistic. Recently, more and more researchers have paid attention to a risk model with dependent claim sizes and/or dependent inter-arrival times, see Chen and Ng [2], Kong and Zong [11], Li et al. [13], Yang and Wang [25], Wang et al. [23], Gao et al. [5], Liu et al. [13], and so on. Therein, Wang et al. [23] have introduced a new dependence structure among r.v.s, which is as follows: Definition 2.1. Say that the r.v.s {ξ n : n 1} are widely upper orthant dependent WUOD, if there exists a finite positive real number sequence {g U n : n 1} such that, for each n 1 and for all

5 FINITE-TIME RUIN PROBABILITY 159 x i,, 1 i n, n n 2.2 P {ξ i > x i } g U n P ξ i > x i. If the inequality 2.2 is changed into n n P {ξ i x i } g L n P ξ i x i, where {g L n : n 1} is another finite positive real number sequence, then say that {ξ n : n 1} are widely lower orthant dependent WLOD. Clearly, if {ξ i, i 1} are WLOD, then { ξ i, i 1} are WUOD. From the definitions of WLOD and WUOD r.v.s, Wang et al. [23] gave the following proposition. Proposition 2.1. i Assume that {ξ i, i 1} are WLOD WUOD r.v.s. If {f i, i 1} are nondecreasing, then {f i ξ i, i 1} are also WLOD WUOD; if {f i, i 1} are nonincreasing, then {f i ξ i, i 1} are WUOD WLOD. ii If {ξ i, i 1} are nonnegative and WUOD r.v.s, then for each n 1, n n E ξ i g U n Eξ i. In particular, if {ξ i, i 1} are WUOD, then for each n 1 and any s >, { n } n E exp s ξ i g U n E exp{sξ i }. For some properties and examples of WUOD and WLOD r.v.s, we refer the readers to Wang et al. [23] and Wang and Cheng [24]. For a nonstandard renewal risk model with WUOD claim sizes and WLOD inter-arrival times, Wang et al. [23] assumed that {X i, i 1}, {Nt, t } and {Ct, t } are mutually independent, and proved

6 151 QINGWU GAO, NA JIN AND HOUCAI SHEN that if F L D, then relation 2.1 holds uniformly for time T varying in a finite interval. In this paper, we will consider the dependence structure among claim sizes from the WUOD structure to a more general dependence structure, whose definition is given below. Definition 2.2. Say that the r.v.s {ξ i, i 1} with distributions V i, i 1, respectively, are pairwise quasi-asymptotically independent if 2.3 P ξ i > x, ξ j > x = ov i x + V j x for i j, i, j 1. Remark 2.1. The term pairwise quasi-asymptotic independence is borrowed from Resnick [17] and Chen and Yuen [3]. Clearly, if {ξ i, i 1} are identically distributed, relation 2.3 is equivalent to P ξ i > x, ξ j > x = op X i > x for i j, i, j 1, that is to say, {ξ i, i 1} are pairwise asymptotically independent or bivariate upper tail independent, see Zhang et al. [27], Gao and Wang [6], and the references therein. We also remark that the pairwise quasiasymptotically independent r.v.s can cover not only common negatively dependent r.v.s but also some positively dependent r.v.s. Inspired by the above results, in this paper we aim at establishing an asymptotic formula 2.1 for the finite-time ruin probability ψ r x, T, < T, in a generalized risk model with constant interest force, where the claim sizes are pairwise quasi-asymptotic independent and their arrival process is an arbitrary counting process. In our main results, we will discuss two cases: one that is the premium process {Ct; t } is independent of {X i, i 1} and {Nt, t }, and the other is that {Ct; t } is not necessarily independent of {X i, i 1} or {Nt, t }. The following are our main results, the first one is concerned with the finite-time ruin probability for any fixed < T <. Theorem 2.1. Consider the generalized risk model introduced in Section 1 with r, where the claim sizes {X i, i 1} are pairwise quasiasymptotically independent r.v.s with common distribution F C, and for any fixed < T < such that ENT >, there exists some

7 FINITE-TIME RUIN PROBABILITY 1511 p > J + F such that ENT p+1 <. Then relation 2.1 holds for the fixed < T <, if one of the following conditions is valid: i the premium process {Ct, t } is independent of {X i, i 1} and {Nt, t }; ii the discounted value of premiums accumulated by time T, defined in 1.2, satisfies that P CT > x = of x. Evidently, for any fixed < T <, the condition that ENT > is equivalent to P NT > = P τ 1 T >, and the condition that ENT p+1 < for some p > J + F, is more relaxed than that of Wang [22], namely, E1 + δ NT < for some δ >. In the next main result, we extend the set for T to an infinite set, ]. Theorem 2.2. Under the conditions of Theorem 2.1 with r > and J F >, we further assume that the claim inter-arrival times {θ i, i 1} are WLOD r.v.s such that 2.4 lim n g Lne ϵn = for some ϵ >, and that the total discounted amount of premiums is finite, that is, 2.5 C = e rs Cds < almost surely. Then relation 2.1 still holds for all < T, if one of the following conditions is valid: i the premium process {Ct, t } is independent of {X i, i 1} and {Nt, t }; ii the total discounted amount of premiums satisfies P C > x = of x. Remark 2.2. In Theorems 2.1 and 2.2 above, condition i has been considered by Wang [22], Yang and Wang [25], Wang et al. [23], Liu et al. [14] and many others; while condition ii, which does not require independence between the premium process and the claim process, allows for a more realistic case that the premium rate varies as a deterministic or stochastic function of the insurer s current reserve, as

8 1512 QINGWU GAO, NA JIN AND HOUCAI SHEN that considered by Petersen [16], Michaud [15], Jasiulewicz [8], Tang [18] and Gao et al. [5]. Applying Theorem 2.1, we now put forward a special case when r =. Corollary 2.1. Consider the above risk model with r =, if the other conditions of Theorem 2.1 are true, then for fixed < T < and any α >, x+αent 2.6 ψ r x, T F xent α 1 F y dy. x The remaining part of the paper is divided into two parts. In Section 3 we present some lemmas that are crucial to proving our main results in Section Lemmas. In order to prove the main results, we need the following lemmas, among which the first lemma is a combination of [1, Proposition 2.2.1] and [2, Lemma 3.5]. Lemma 3.1. If a distribution V C, then: i for any p > J + V, there are positive constants C and D such that V y V x C x/yp holds for all x y D; and, for any p < J V, there are positive constants Ĉ and D such that V y V x Ĉ x/yˆp holds for all x y D; ii for any p > J + V, the following holds: 3.3 x p = ov x.

9 FINITE-TIME RUIN PROBABILITY 1513 The following second lemma will play an important role in proving the main results and is also of its own merit and close to the spirit of [21, Proposition 5.1]. Lemma 3.2. If {ξ i, 1 i n} are n real-valued and pairwise quasiasymptotically independent r.v.s with distributions V i C, 1 i n, respectively, such that 3.4 V i x = ov i x, 1 i n, then for any fixed < a b <, n 3.5 P c i ξ i > x n P c i ξ i > x holds uniformly for all c n = c 1, c 2,..., c n [a, b] n, that is, lim sup c n [a,b] n n P c iξ i > x n P c iξ i > x 1 =. Proof. Relation 3.5 is clear if n = 1. Hence, we assume n 2. On the one hand, for an arbitrarily fixed v 1/2, 1, 3.6 n P n c i ξ i > x P {c i ξ i > vx} n + P c i ξ i > x, = I 1 + I 2. n {c i ξ i vx} For I 1, by V i C, 1 i n, and the arbitrariness of v 1/2, 1, we have 3.7 lim v 1 lim sup I sup 1 n c n [a,b] n P c iξ i > x lim lim sup v 1 sup c n [a,b] n n V ivx/c i n V ix/c i = 1.

10 1514 QINGWU GAO, NA JIN AND HOUCAI SHEN For I 2, it follows that 3.8 n I 2 = P c i ξ i > x, n n P k=1 n n k=1,i k,i k P c i ξ i > x n < max c kξ k vx 1 k n c i ξ i > 1 vx, c k ξ k > x n 1 vx n 1, c kξ k > x. n Since 1/n > 1 v/n 1, we obtain from 2.3, 3.8 and V i C D, 1 i n, that, for all c n [a, b] n, 3.9 lim n P c iξ i > x lim I 2 n n k=1,i k P c iξ i > [1 vx]/n 1, c k ξ k > [1 vx]/n 1 P c i ξ i > x + P c k ξ k > x n n lim k=1,i k P ξ i > 1 v/n 1x/b, ξ k > 1 v/n 1x/b V i [1 v]/[n 1]x/b + V k [1 v]/[n 1]x/b Vi[1 v]/[n 1]x/b + V k [1 v]/[n 1]x/b V i x/a + V k x/a =. Thus, substituting 3.7 and 3.9 into 3.6 yields that n 3.1 P c i ξ i > x n P c i ξ i > x holds uniformly for all c n [a, b] n.

11 FINITE-TIME RUIN PROBABILITY 1515 On the other hand, for an arbitrarily fixed w > 1, 3.11 n P For I 3, it holds that 3.12 I 3 = n k=1 n c i ξ i > x P c i ξ i > x, max c kξ k > wx 1 k n n n P c i ξ i > x, c k ξ k > wx k=1 P c k ξ k > wx, 1 i<j n = I 3 I 4. n P c k ξ k > wx k=1 n,i k n n P c k ξ k > wx I 5. k=1 P c i ξ i > wx, c j ξ j > wx c i ξ i > 1 wx n k=1,i k P c i ξ i < 1 wx n 1 By 3.4 and V i C D, 1 i n, we have that, for all c n [a, b] n, lim sup I 5 n P c iξ i > x lim sup n n k=1,i k lim sup n P ξ i > w 1/n 1x/b P ξ i > x/a This, along with 3.12, leads to n k=1 i k P c i ξ i < 1 wx/n 1 P c i ξ i > x P ξ i < 1 w/n 1x/b P ξ i > w 1/n 1x/b = lim inf I 3 n P c lim inf iξ i > x n k=1 P c kξ k > wx n k=1 P c kξ k > x.

12 1516 QINGWU GAO, NA JIN AND HOUCAI SHEN For I 4, by 2.3 we see that, for all c n [a, b] n, 3.14 lim sup I 4 n P c iξ i > x lim sup lim sup 1 i<j n 1 i<j n Viwx/b + V j wx/b V i wx/a + V j wx/a =. P c i ξ i > wx, c j ξ j > wx P c i ξ i > wx + P c j ξ j > wx P ξ i > wx/b, ξ j > wx/b V i wx/b + V j wx/b From 3.11 to 3.14, it follows that lim w 1 lim inf n P inf c iξ i > x n c n [a,b] n P c iξ i > x lim lim inf w 1 which implies that n 3.15 P c i ξ i > x holds uniformly for all c n [a, b] n. inf c n [a,b] n n P c i ξ i > x n V iwx/c i n V ix/c i = 1, Consequently, we will complete the proof of this lemma by combining 3.1 and Remark 3.3. Clearly, if {ξ i, 1 i n} are nonnegative and pairwise quasi-asymptotically independent, then condition 3.4 is satisfied naturally, and it can be canceled for Lemma 3.2. Lemma 3.3. Consider the counting process {Nt, t } in 1.5 with WLOD inter-arrival times {θ i, i 1} satisfying 2.4. Then, for any fixed T > and any p >, 3.16 ENT p <.

13 FINITE-TIME RUIN PROBABILITY 1517 Proof. Note that, by the Markov inequality and Proposition 2.1, 3.17 ENT p n p P τ n T e T e T n p Ee τn g L nn p exp{n logee θ 1 }. Applying 2.4 and setting ϵ = logee θ 1 c for some c >, we can find some n such that, for all n n, g L n e cn exp{ n logee θ 1 }, which, along with 3.17, leads to n 1 ENT p e T This ends the proof. g L nn p Ee θ 1 n + n p e cn n n <. The lemma below is due to [22, Lemma 3.5]. Lemma 3.4. For the generalized risk model introduced in Section 1 with ENT >, for any fixed < T <, we have P X i e rτi 1 {τi T } > x = F xe rt dent. 4. Proofs of main results. Proof of Theorem 2.1. We now proceed to prove Theorem 2.1, and the ideas are inspired by the proof of [22, Theorem 2.2]. Recalling the surplus process 1.1, we attain its discounted value as Nt 4.1 Ũ r t = e rt U r t = x + Ct X i e rτi, t,

14 1518 QINGWU GAO, NA JIN AND HOUCAI SHEN where Ct is defined by 1.2. By the definition 1.3 of the finite-time ruin probability, we have ψ r x, T = P Ũrt < for some < t T Nt = P X i e rτ i > x + Ct for some < t T. Then, it follows that NT 4.2 ψ r x, T P X i e rτi and 4.3 ψ r x, T = P <t T { Nt X i e rτi > x NT P X i e rτi > x + CT. } > x + Ct According to the conditions of Theorem 2.1, we conclude that, for any given ε > and any fixed T >, there exists a positive integer m = m T, ε > 1 such that 4.4 Ce rt p ENT p+1 1 {NT >m} ENT ε, where C > and p > J + F 3.1. are two constants as similar to?that in On the one hand, we deal with 4.2 to show the upper bound of ψ r x, T. Let m be fixed as above. It is easy to see that 4.5 NT P X i e rτi = m + > x n=m +1 n P X i e rτi > x, NT = n = H 1 + H 2.

15 FINITE-TIME RUIN PROBABILITY 1519 First consider H 1. Let Gt 1, t 2,..., t n+1 be the joint distribution of the random vector τ 1, τ 2,..., τ n+1, where n = 1, 2,..., m. Thus, by Lemma 3.2 and the independence of {X i, i 1} and {Nt, t }, we derive that there exists an x 1 = x 1 T, ε such that, for all x > x 1, 4.6 H 1 = m {<t 1 t 2. t n T,t n+1>t } n P X i e rt i > x dgt 1, t 2,..., t n+1 m 1 + ε n {<t 1 t 2 t n T,t n+1 >T } P X i e rt i > x dgt 1, t 2,..., t n+1 m n = 1 + ε P X i e rτ i > x, NT = n 1 + ε = 1 + ε = 1 + ε n=i P X i e rτ i > x, NT = n P X i e rτi 1 τi T > x F xe rt dent, where, in the last step, we used Lemma 3.4. Next consider H 2, which is divided into two parts as: 4.7 H 2 = m <n<x/d = H 21 + H 22, + n x/d n P X i e rτ i > x, NT = n where the constant D is the same one as in 3.1. For H 21, for the fixed T >, it follows from 3.1 and 4.4 that, for all x D, 4.8 H 21 m <n<x/d n P X i > x P NT = n

16 152 QINGWU GAO, NA JIN AND HOUCAI SHEN m <n<x/d CF x nf x/np NT = n m <n<x/d n p+1 P NT = n CF xent p+1 1 {NT >m} εf x ENT Cε e rt p F xe rt dent. For H 22, by 3.3 and 4.4, there exists an x 2 = x 2 ε such that, for all x > max{d, x 2 }, 4.9 H 22 P NT x/d x/d p 1 ENT p+1 1 {NT >m} ε F xent p+1 1 {NT >m} ε 2 F xe rt dent. Therefore, we obtain from that, for all x>x 3 = max{d,x 1,x 2 }, 4.1 ψ r x, T 1 + ε + Cε + ε 2 F xe rt dent. On the other hand, we now deal with 4.3 to estimate the lower bound of ψ r x, T. Under Theorem 2.1 i, by conditioning on CT defined in 1.2 and along similar lines to the derivation of H 1, there exists an x 4 = x 4 T, ε such that, for all x > x 4, 4.11 ψ r x, T m {<t 1 t 2 t n T,t n+1>t } P CT dy dgt 1, t 2,..., t n+1 m n 1 ε n P X i e rt i {<t 1 t 2 t n T,t n+1 >T } > x + y

17 FINITE-TIME RUIN PROBABILITY 1521 P X i > xe rti + ye rt P CT dy dgt 1, t 2,..., t n+1 m n 1 ε 2 P X i e rτi > x, NT = n = 1 ε 2 n=m +1 n P X i e rτi > x, NT = n = 1 ε 2 F xe rt dent 1 ε 2 H 3, where m is defined in 4.4, and the third step is from the dominated convergence theorem and F C L. For H 3, we obtain from 3.1 and 4.4 that, for all x D, H 3 n=m +1 = F x n P X i > x, NT = n n=m +1 np NT = n = F xent 1 {NT >m } εf x ENT Ce rt p ε F xe rt dent, which, together with 4.11, implies that, for all x x 5 = max{x 4, D}, 4.12 ψ r x, T 1 ε 2 F xe rt dent 1 ε 2 ε F xe rt dent = 1 ε 3 F xe rt dent. Therefore, we conclude from 4.1 and 4.12 that, for all x max{x 3, x 5 }, ε 3 F xe rt dent ψ r x, T

18 1522 QINGWU GAO, NA JIN AND HOUCAI SHEN 1 + ε + Cε + ε 2 F xe rt dent. Under Theorem 2.1 ii, we see from F C that, for each t i >, i = 1, 2,..., n, appearing in 4.6, F 1 + lxe rt i lim lim inf = 1, l F xe rti which means that there exist an l > and an x 6 = x 6 ε, t i such that, for all x x 6, 4.14 F 1 + l xe rt i 1 εf xe rt i. It follows from 4.3 that, for l > as above, 4.15 NT ψ r x, T P X i e rτi P P NT NT = H 4 H 5. X i e rτ i > x + CT, CT l x > 1 + l x, CT l x X i e rτ i > 1 + l x P CT > l x For H 4, arguing as 4.11 and H 3 and using 4.14 can yield that, for all x max{x 5, x 6 }, 4.16 H 4 m {<t 1 t 2. t n T,t n+1>t } n P X i e rt i > 1 + l x dgt 1, t 2,..., t n+1 m 1 ε n {<t 1 t 2 t n T,t n+1>t } P X i > 1 + l xe rt i dgt 1, t 2,..., t n+1

19 FINITE-TIME RUIN PROBABILITY 1523 m 1 ε 2 1 ε 3 n P X i e rτi > x, NT = n F xe rt dent. For H 5, by Theorem 2.1 ii and F C D, we get lim sup H 5 = lim sup F x P CT > l x F l x F l x F x =, from which we derive that there exists an x 7 = x 7 ε such that, for all x max{x 7, D}, 4.17 H 5 εf x C ε F xe rt dent, where C = Ce rt p /ENT. Substituting 4.16 and 4.17 into 4.15, we have that, for all x x 8 = max{x 5, x 6, x 7 }, 4.18 ψ r x, T 1 ε 3 C ε Hence, for all x max{x 3, x 8 }, it holds that 4.19 F xe rt dent. 1 ε 3 C ε F xe rt dent ψ r x, T 1 + ε + Cε + ε 2 F xe rt dent. As a result, by using 4.13 and 4.19, and taking into account the arbitrariness of ε >, we prove that, for any fixed < T <, relation 2.1 holds under either of conditions i and ii of Theorem 2.1. Proof of Theorem 2.2. When < T <, we know from 2.4 and Lemma 3.3 that Theorem 2.2 is a special case of Theorem 2.1, and then it follows immediately from the proof of Theorem 2.1. Hence, in the rest, we only need to prove the case when T =. By 1.4 and 4.1, we have ψ r x, = P Ũrt < for some t <.

20 1524 QINGWU GAO, NA JIN AND HOUCAI SHEN Thus, we see that 4.2 P X i e rτi > x+ C ψ r x, P X i e rτi > x, where C is defined by 2.5. According to [26, Remark 2], we find that, if F C with J F > and r >, then 4.21 P X i e rτi > x P X i e rτi > x = F xe rt dent. This and 4.2 show that, for any given ε >, there exists an x 9 = x 9 ε such that, for all x x 9, 4.22 ψ r x, 1 + ε F xe rt dent. Subsequently, we will analyze the lower bound of ψ r x,. For any fixed p and p, p > J + F, p < J F, we apply 3.1 and 3.2 to derive that, for all x max{d, D} and all < T <, 4.23 T F xert dent F xert dent = T F xert /F x dent F xert /F x dent C Ĉ T e r ˆpt dent e rpt dent. By Proposition 2.1, it follows that e r ˆpt dent = e r ˆpt dp τ n t = Ee r ˆpτn g L n exp{n logee r ˆpθ 1 }. Setting ϵ = logee r ˆpθ 1 c in 2.4 for some c > proves that, there exists some n such that, for all n n, Hence, we get that g L n e cn exp{ n logee r ˆpθ 1 } e r ˆpt dent n 1 g L nee r ˆpθ1 n + n n e cn <.

21 FINITE-TIME RUIN PROBABILITY 1525 Likewise, we also get 4.25 e rpt dent <. Consequently, combining 4.23, 4.24 and 4.25, the right-hand side of 4.23 tends to as T tends, which yields that, for the given ε >, there exists a < T < such that 4.26 T F xe rt dent ε F xe rt dent holds for all x max{d, D}. Under Theorem 2.2 i, by applying 4.2, conditioning on C and arguing as in 4.11 and 4.12, we obtain that there exists an x 1 = max{ D, x 5 } such that, for all x x 1 and for the fixed < T < as that in 4.26, 4.27 NT ψ r x, P X i e rτi > x + C 1 ε 3 F xe rt dent = 1 ε 3 F xe rt dent T = 1 ε 4 F xe rt dent, where, in the last step, we used the inequality Thus, combining 4.22 and 4.27, we derive that, for all x max{x 9, x 1 }, ε 4 F xe rt dent ψ r x, 1+ε F xe rt dent. Under Theorem 2.2 ii, by using 4.2 and arguing as in the derivation of 4.15, we attain that for l > and < T < as in 4.14 and 4.26, respectively, 4.29 NT ψ r x, P X i e rτi > 1 + l x P C > l x = H 6 H 7.

22 1526 QINGWU GAO, NA JIN AND HOUCAI SHEN For H 6, along the same lines of H 4 with T replaced by T can imply that, for all x x 11 = max{x 6, x 1 }, 4.3 H 6 1 ε 3 F xe rt dent 1 ε 4 F xe rt dent, where the last step is from For H 7, by a derivation similar to 4.17, Theorem 2.2 ii and F C D, there exists an x 12 = x 12 ε such that, for all x x 12, 4.31 H 7 εf x C ε F xe rt dent, where C is the same as that in So, from , it follows that, for all x max{x 11, x 12 }, ψ r x, 1 ε 4 C ε F xe rt dent. This, along with 4.22, can show that, for all x max{x 9, x 11, x 12 }, ε 4 C ε F xe rt dent ψ r x, 1 + ε F xe rt dent. Therefore, combining 4.28 with 4.32 and using the arbitrariness of ε >, we prove that, for T =, relation 2.1 holds under either of conditions i and ii of Theorem 2.2, and this ends the proof. Proof of Corollary 2.1. Clearly, by Theorem 2.1, we obtain that, for any fixed < T <, Note that, for fixed < T <, ψ r x, T F xent. x+αent F xent α 1 F y dy x F x + αent ENT F xent,

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