ASYMPTOTICS FOR RUIN PROBABILITIES OF A NON-STANDARD RENEWAL RISK MODEL WITH DEPENDENCE STRUCTURES AND EXPONENTIAL L´EVY PROCESS INVESTMENT RETURNS

. Consider a non-standard renewal risk model with dependence structures, where claim sizes follow a one-sided linear process with independent and identically distributed step sizes, the step sizes and inter-arrival times respectively form a sequence of independent and identically distributed random pairs, with each pair obeying a dependence structure. An insurance company is allowed to make risk-free and risky investments, where the price process of the investment portfolio follows an exponential L´evy process. When the step-size distribution is dominatedly-varying-tailed, some asymptotic estimates for the ﬁnite-and inﬁnite-time ruin probabilities are obtained.

1. Introduction. It is well known that the standard renewal risk model (proposed by Sparre Andersen in 1957) has played a fundamental role in risk theory. This standard framework is based on many independence assumptions; e.g., the claim sizes, X n , n ≥ 1, and the inter-arrival times Y n , n ≥ 1, respectively form a sequence of independent and identically distributed (i.i.d.) random variables, and the two sequences are mutually independent. It is worth pointing out that these independence assumptions are mainly for mathematical tractability rather than practical relevance. Moreover, these independence assumptions make the renewal risk model too restrictive for practical problems. Therefore, recently, more and more researchers have started to propose different non-standard extensions to the 156 JIANGYAN PENG AND DINGCHENG WANG standard renewal risk model with various dependence structures. See [1], [2], [4], [5], [7], [13], [16], [25], [28], among others.
In this paper, we improve the model through introducing suitable dependence structures among claim sizes, and between claim sizes and inter-arrival times. On the one hand, with the increasing complexity of insurance and reinsurance products, researchers have been paying an increasing amount of attentions to the modeling of dependent risks. Throughout this paper, we make an assumption on the dependence of claim sizes {X n , n ≥ 1} as follows.
As pointed out by a large amount of literature ( [9], [27], [37], among others), linear processes including the ARMA model and the fractional ARIMA model, are often used in time series analysis, and are also a type of widely used dependence structures in insurance risk theory.
On the other hand, if the deductible retained to the insured is raised, then the inter-arrival time will increase and the claim sizes would decrease because small losses will be ruled out and retained by the insured. Hence, the independence between the claim size X n and the inter-arrival time Y n is unreasonable in many insurance applications. During the last few years, the non-standard renewal risk models with dependence structures between X n and Y n have attracted an increasing amount of attentions from researchers in risk theory. See [2], [3], [5], [7], [13], [16], [24] and [25].
Motivated by [39], we make another type of assumption on the dependence structure of (ε n , Y n ) as follows.
Consider the non-standard renewal risk model with Assumptions 1.1 and 1.2, in which the inter-arrival times {Y n , n ≥ 1}, not independent of the claim sizes {X n , n ≥ 1}, form a sequence of i.i.d., nonnegative and non-degenerate at-zero r.v.'s with common distribution function G. Then, the arrival times of the successive claims τ n = n i=1 Y i , n ≥ 1, constitute a renewal counting process where I A denotes the indicator function of an event A, and by convention, τ 0 = 0. Denote the renewal function of {N (t), t ≥ 0} as The aggregate claims up to time t (≥ 0) are given by the compound sum with S t = 0 when N (t) = 0. Suppose that an insurer is allowed to make risk-free and risky investments. Based on a amount of empirical investigations of stock markets indicating that the price processes of many stocks have sudden downward or upward jumps which cannot be explained by a continuous geometric Brownian motion, we consider a more general exponential (also known as geometric) Lévy process with jumps to model the price process of the investment portfolio. The price processes of the risk-free and risky assets, respectively, satisfy Z 0 (t) = e rt and Z 1 (t) = e L(t) , t ≥ 0, where r > 0 is the risk-free rate, and the process {L(t), t ≥ 0} is a Lévy process. That is, L(0) = 0, {L(t), t ≥ 0} has independent and stationary increments, is stochastically continuous, and is right continuous with left limit (see [31] for general theory of Lévy process). Let (γ, σ 2 , ν) be the characteristic triplet of a Lévy process {L(t), t ≥ 0}, where γ ∈ R, σ ≥ 0 are two constants and ν is a Lévy measure satisfying ν({0}) = 0 and where x ≥ 0 is the initial surplus of the insurance company and c > 0 is the constant premium rate. Under the conditions of (1) and (2), we remark that the U θ (t) is a generalization of the surplus processes considered by [10], [17], [21], [30], [40] and [41]. Define the Laplace exponent of the process {L θ (t), t ≥ 0} as If ψ θ (s) < ∞, then By the proof of Lemma 4.1 in [22], we can get that ψ θ (s) < ∞ for all θ ∈ (0, 1) and s ≥ 0, and if 0 < EL(1) < ∞ and either σ > 0 or v((−∞, 0)) > 0, then there exists a unique positive κ θ > 0 such that ψ θ (κ θ ) = 0. Thus, if 0 < EL(1) < ∞, either σ > 0 or ν((−∞, 0)) > 0, then the convexity of ψ θ (·) with ψ θ (0) = 0 and ψ θ (0) = −EL θ (1) < 0 implies that for any fixed θ ∈ (0, 1), As usual, define the finite-time ruin probability and the infinite-time ruin probability of the non-standard renewal risk model (12), respectively, as and The main goal of this paper is to investigate ruin probabilities of the non-standard renewal risk model (12) with stochastic investment returns under Assumptions (1) and (2), and examine how tail probabilities are influenced by the dependence structures. When the common distribution function of the step sizes in (1) is heavy tailed, we establish some asymptotic formulas that hold for finite or infinite time horizons.
There are a few recent articles that are related to our study on asymptotic estimates for ruin probabilities of non-standard renewal risk models with stochastic investment returns. [17], [18] and [38] considered the similar problem, but their works were concentrated on the one-sided linear, bivariate upper tail independent, and upper tail asymptotic independence (UTAI) claim-size models (without dependence structures between the claim sizes and the inter-arrival times), respectively. [16] used the dependence structure as proposed by [3] to characterize the relation between the claim sizes and the inter-arrival times (often termed as a timedependent renewal risk model). [24] studied the uniformly asymptotic tail behavior for a time-dependent renewal risk model with stochastic return, under a dependence structure between the extended-regularly-varying tailed claim sizes and the inter-arrival times. But for the time-dependent renewal risk model, [16] and [24] considered the case of independent claim sizes. However, we not only use a different and meaningful bivariate Sarmanov dependent structure to characterize the dependence relation between claim sizes and inter-arrival times, but also adopt an interesting one-sided linear process to model the dependent claim sizes. Hence, our obtained results partly extend the results of the above papers. Until now, no paper has simultaneously addressed continuous-time renewal risk models with stochastic investment returns under dependent structures between claim sizes and inter-arrival times, and among claim sizes.
The rest of this paper consists of three sections. Section 2 introduces some notations and states the main results of the paper, Section 3 provides some important lemmas, and Section 4 proves the main results.
2. Notations and main results. Throughout this paper, C represents a generic positive constant, which may vary with the context. Hereafter, all limit relations are for x → ∞ unless stated otherwise. For two positive functions a(·) and b(·) satisfying In finance and insurance, heavy tailed random variables play an important role in modeling extremal events, since they can model jumps and jumbo claims realistically. Now we recall some related classes of heavy-tailed distribution functions. In this paper, we focus on the so-called dominated variation class, denoted by D. By definition, a distribution F on [0, ∞) belongs to the class D, denoted by F ∈ D, if lim sup < ∞, for any y > 0.
We also recall some other classes of heavy-tailed distribution functions which are crucial for our purpose. We say that a d.f. F [0, ∞) belongs to the class C (has a consistently varying tail), denoted by F ∈ C, if We say that a d.f. F [0, ∞) belongs to the extended regular variation class (ERV), denoted by F ∈ ERV(−α, −β), if there are some 0 < α ≤ β < ∞ such that the relation We say a d.f. F has a regularly varying tail with tail index −α < 0, denoted by = y −α , for any y > 0.
We say that a d.f. F [0, ∞) belongs to the class L (is long-tailed), denoted by F (x) = 1, for any (or, equivalently, for some) y = 0.
It is well known that R ⊂ ERV ⊂ C ⊂ D ∩ L ⊂ D, D L and L D.
For more details of heavy-tailed distributions, see [14]. Now we need two significant indices of a general distribution function F following [33]. For any y > 0, we set and then where J + F and J − F are called the upper and lower Matuszewska indices, respectively.
Here, we only consider the case θ ∈ [0, 1), since the insurer is not allowed to invest all his or her wealth into risky assets. Now we begin to state our main results.
Next we discuss some special cases of Theorem 2.2.
Remark 2.1. By letting ϕ 0 = 1, ϕ n = 0 for all n ≥ 1 in (1), and 1+πφ (2), respectively, we can see that the non-standard renewal risk model (12) includes two special cases that the claim sizes are i.i.d, and ε is independent of Y . Thus, Theorem 2.1 partly extends the results of [20], in which the claim sizes are i.i.d, and the claim sizes, X n , n = 1, 2, . . ., and the inter-arrival times Y n , n = 1, 2, . . ., are mutually independent. Theorem 2.3 and Corollary 2.2 partly extend the results of [17] and [18], in which ε is considered to be independent of Y .
Remark 2.2. Consider the case that step sizes of claims have regularly varying tails. Theorem 2.3 and Corollary 2.2 show that in finite and infinite times, the extreme of insurance risk always dominates the extreme of the financial risk because the tail probability of step sizes determines the exact decay rate of the ruin probabilities. However, for the case that the claim sizes are i.i.d., Theorem 4.4 in [22] shows for the case of dangerous investment (ψ θ (α) > 0 since α > κ θ ) that the extreme of the financial risk finally dominates the extreme of the insurance risk when the claim sizes have regularly varying tails.
3. Lemmas. In this section, we need a series of lemmas to prove our main results. First, we begin with some properties of the class D in the lemma below, which is due to [6], [11] and [33].
Lemma 3.1. If a distribution F ∈ D, then for any p > J + F , we have and F ∈ D if and only if L F > 0.
Lemma 3.2. Let X and Y be two independent and nonnegative random variables, where X is distributed by F ∈ D and Y is nonnegative and nondegenerate at 0 satisfying EY p < ∞ for some p > J + F . Then, the distribution of the product XY belongs to the class D and P {XY > x} F (x).
Proof. See Theorem 3.3 (iv) of [11] (also Lemma 3.8 of [33]). Lemma 3.3. Let ε and Θ be two independent and nonnegative random variables with ε distributed by F . Then, we have the following two results. (i) If F ∈ D, then for arbitrarily fixed δ > 0 and J + F < p 2 < ∞, there exists a positive constant C without relation to Θ and δ such that for all large x, Proof. See Lemma 3.2 in [20].
there exists a positive constant C without relation to Θ and δ such that for all large x, Proof. See Lemma 4.1.5 in [35].
Proof. See the proof of sufficiency in [42] and the proof of the necessity in [32]. Then we have: Proof. See Proposition 2.1 and Lemma 2.1 of [26], respectively.
In order to facilitate subsequent expression, we denote 164 JIANGYAN PENG AND DINGCHENG WANG The following inequalities will play crucial roles in the proof of Theorem 2.1. For 0 < p ≤ 1, by C r inequality and sup n≥0 ϕ n < ∞, we have for any 0 ≤ s ≤ T and For p > 1, by Hölder's inequality and C r inequality, we have for any 0 ≤ s ≤ T and i ≥ 1, Lemma 4.1 in [22] has shown that ψ θ (s) < ∞ for all θ ∈ (0, 1) and s ≥ 0. Especially, (7) and (13) imply ψ 0 (p) = −rp < 0. Since there exists some h > 0 such that E[e hN (T ) ] < ∞ by Lemma 3.2 in [19], we can obtain that, Then, we can get for any p > 0, 0 ≤ s ≤ T, and i ≥ 1, Hence, we have for any δ > 0, any p > 0 and any q ≥ 0, The following lemma will play a crucial role in the proof of the main results. and hold for all 0 < T < ∞ with P {τ 1 ≤ T } > 0, and the distribution function of ε i ϑ i (T ) belongs to the class D.
Proof. By the conditions of Theorem 2.1, let lim x→∞ φ 1 (x) = d 1 , and there exist two constants b 1 > 1 and b 2 > 1 such that |φ 1 (x)| ≤ b 1 − 1| and |φ 2 (y)| ≤ b 2 − 1 for all x ∈ D ε and y ∈ D Y . Obviously, d 1 < b 1 . According to the definitions of bivariate Sarmanov distribution (2) and the distributionsF andG in (28), we have for every where , t ≥ 0} are mutually independent, and are also independent of {ε i , i ≥ 1} and {Y i , i ≥ 1}. Since lim x→∞ φ 1 (x) = d 1 , and by relation (28), we havẽ Hence, since F (x) ∈ D, it follows that As for the first term on the right-hand side of (39), since Y * i is identically distributed to Y and independent of {L θ (t), t ≥ 0} and {Y k , k ≥ 1}, relation (36) implies that for some p > J + F and every i ≥ 1, By relation (41) and independence among ε * i , Y * i , {Y n , n ≥ 1} and {L θ (t), t ≥ 0}, Lemma 3.2 implies that the distribution of ε * i ϑ i (Y * i , T ) belongs to the class D and Because of independence amongỸ * i , {Y n , n ≥ 1} and {L θ (t), t ≥ 0}, relations |φ 2 (y)| ≤ b 2 − 1 and (28), we get Similar to the proof of relation (36), the aforementioned inequality implies that for any δ > 0, any p > 0 and any q ≥ 0, For each i ≥ 1, similar to the proof of the distribution of ε * Next we use the approach in [39], but there are many changes in this proof because the coefficients {ϕ n , n ≥ 1} bring much trouble. By Chebyshevs inequality, (33), (41) and (42), we can obtain for some p > J + F , According to the fact that the distribution of ε * i ϑ i (Y * i , T ) belongs to the class D, Lemma 3.4 implies that there exists a positive functiong(·) such that
Proof. Obviously, let g(x) = x/ ln x which satisfies the assumption condition. Take p * > J + F + ε for some ε > 0. By Chebyshevs inequality, (33) and (36), we can obtain for every i ≥ 1, By the same method as the proof of the first relation, we can prove the second relation together with relation (47). This ends the proof of Lemma 3.7.
Proof. Let g(x) be the function defined in Lemma 3.7. According to Lemma 3.7, we have for some p > J + F and 1 ≤ i < j, As in the proof of (39), we split the probability P {ε i ϑ i (T ) > x, ε j > x/g(x)} on the right-hand side of the aforementioned inequality into four parts as As for I 1 , by the fact that , t ≥ 0} are mutually independent, and are also independent of {ε i , i ≥ 1} and {Y i , i ≥ 1}, Lemma 3.6 implies that where the last inequality follows from (47) and lemma 3.2.
Similarly, according to (28) and (40), we have Together with (38), we complete the proof of the lemma.
Proof. Firstly, we give the asymptotic lower bound of (56). For every integer N such that As for J 1 (x, N ), by (34) in Lemma 3.3 and (36), it can be shown that for some p > J + F , all large x and sufficiently large N , Similarly, according to (28), (40) and the boundness of φ 2 (y), we obtain that as first x → ∞ and then N → ∞, Hence, (38) implies that as first x → ∞ and then N → ∞, This means that for any 0 < δ < 1, there exist sufficiently large N 0 and x 1 such that for all x ≥ x 1 , and By applying Lemma 3.9, it holds that for the aforementioned large N 0 , This, together with (59), implies that, which, by the arbitrariness of δ, gives the asymptotic lower bound.
Then we proceed to the asymptotic upper bound. For any 0 < l < 1 and N 1 ≥ 1, we have For any 0 < ρ < 1, we have Similar to the proof of the asymptotic upper bound for any fixed N in Lemma 3.9, by letting lρ → 1(l → 1), we can obtain that Also, similar to (58), first letting x → ∞ and then N 1 → ∞, This completes the proof of Lemma 3.10.
Proof. It follows from Lemma 3.6 that for any fixed N ≥ 1, Similar to the proof of (57), it follows from (34) in Lemma 3.3 and (47) that for . Then, (60) holds that as first x → ∞ and then N → ∞. Hence, this, together with (59) and (62), implies that (61) holds for large x.
Now we turn to the proof of Theorem 2.1.
Hence, the proof of Theorem 2.1 is completed.
Firstly, we deal with the following inequalities, which are crucial to the proof of Theorem 2.2.
The remaining results in Lemma 3.6 also hold for T = ∞. Hence, Lemma 3.6 also holds for T = ∞ under the conditions of Theorem 2.2. Take J + F < p < κ θ . By (68) and (69), the results in Lemmas 3.7-3.9 also hold for T = ∞ under the conditons of Theorem 2.2.
The remaining results in Lemma 3.10 also hold for T = ∞. Hence, relation (56) also holds for T = ∞ under the conditions of Theorem 2.2. Finally, we consider that (60) and (61) in Lemma 3.11 also hold for T = ∞ under the conditions of Theorem 2.2. As for (60), it follows from (35) in Lemma 3.3 and (69) that for any fixed p 1 > 0 and p 2 > 0 such that 0 < p 1 < α ≤ J − F ≤ J + F ≤ β < p 2 < κ θ ≤ ∞, all large x and sufficiently large N , = o(F (x)).

RUIN PROBABILITIES WITH DEPENDENCE STRUCTURES 183
The remaining results in Lemma 3.11 also hold for T = ∞. Hence, relation (61) also holds for T = ∞ under the conditions of Theorem 2.2. By the above results, we immediately obtain relation (30) also holds for T = ∞ under the conditions of Theorem 2.2.