Infinite-time ruin probability of a renewal risk model with exponential Levy process investment and dependent claims and inter-arrival times

We investigate the infinite-time ruin probability of a renewal risk model with exponential Levy process investment and dependent claims and inter-arrival times. Assume that claims and corresponding inter-arrival times form a sequence of independent and identically distributed copies of a random pair \begin{document} $(X,T)$ \end{document} with dependent components. When the product of the claims and the discount factors of the corresponding inter-arrival times are heavy tailed, we establish an asymptotic formula for the infinite-time ruin probability without any restriction on the dependence structure of \begin{document} $(X,T)$ \end{document} .


1.
Introduction. Consider a renewal risk model in which claims, {X n } n≥1 , constitute a sequence of independent, identically distributed (i.i.d.), and positive random variables (r.v.s) with generic random variable (r.v.) X, common distribution F such that F (x) = 1 − F (x) > 0 for all x > 0, and their arrival times {τ n } n≥1 constitute a renewal counting process where 1 A denotes the indicator function of the set A. For later use, we write τ 0 = 0. To avoid triviality, we assume that τ 1 is a nonnegative and non-degenerate at 0. Denote the renewal function of {N (t)} t≥0 as P{τ n ≤ t}.
The inter-arrival times {T n } n≥1 , T n = τ n − τ n−1 , form a sequence of i.i.d. and positive r.v.s with generic r.v. T . Hence, the total amount of claims up to time t ≥ 0 can be written as X n with S(t) = 0 when N (t) = 0. The total amount of premiums accumulated up to t ≥ 0 is denoted by C(t) = ct, where c is a positive constant.
The complete independence of {X n } n≥1 and {T n } n≥1 is far unrealistic with the increasing complexity of economic environment. For example, 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, many researchers introduced dependence structures between {X n } n≥1 and {T n } n≥1 into renewal risk models. We refer readers to Asimit and Badescu [1], Li et al. [14], Li [15], Fu and Ng [9], among many others. Motivated by this, we introduce a type of dependence structure as follows: Assumption 1. {(X n , T n )} n≥1 are i.i.d. copies of (X, T ) with dependent components.
Assume that an insurer can make investment in risk-free and risky assets. The price processes of the risk-free and risky assets, respectively, satisfy R 0 (t) = e rt and R 1 (t) = e L(t) , t > 0, where r > 0 is the risk-free interest rate, {L(t)} t≥0 is a Lévy process. That is to say, L(0) = 0, {L(t)} t≥0 has independent and stationary increments, and is right continuous with left limit. Let (γ, σ 2 , ν) be the characteristic triple of {L(t)} t≥0 , where γ ∈ R, σ ≥ 0 and Lévy measure ν satisfies ν(0) = 0 and R (x 2 ∧1)ν(dx) < ∞. For the general theory of Lévy processes, see Sato [17] and Cont and Tankov [6]. Suppose that the insurer continuously invests a constant fraction θ ∈ (0, 1) of its surplus in the risky asset and invests the remaining surplus in the risk-free asset (see e.g. Emmer et al. [7] and Emmer and Klüppelberg [8]). This strategy is classical in financial portfolio optimization (see Korn [13], Section 2.1). The fraction θ is called the investment strategy.
Although insurance processes and investment processes may be weakly dependent in economic environment, we assume that they are independent, which allows for a very explicit analysis of the integrated risk process. We refer reader to Klüppelberg and Kostadinova [12]. Thus, we have: Then, by Lemma 2.2 of Klüppelberg and Kostadinova [12], we can obtain the integrated risk process (IRP) where x > 0 is the initial surplus of the insurer. Denote the discounted net loss process by Now we can define the infinite-time ruin probability of IRP (2) by By Assumption 1, X n and e L θ (τn−1)−L θ (τn) are dependent. Without the consideration of specific dependence structures, we make the following assumption:

denoted by H, belongs to the consistent variation class (C).
Trivially, if X n is independent of T n and has a distribution belonging to C, we can derive that for Then, Lemma 3.3 shows that H belongs to C. If (X n , T n ) follows a bivariate Farlie-Gumbel-Morgenstern (FGM) distribution, i.e., P{X n ≤ x, T n ≤ y} = P{X n ≤ x}P{T n ≤ y}(1 + ϑP{X n > x}P{T n > y}), |ϑ| < 1.

RONGFEI LIU, DINGCHENG WANG AND JIANGYAN PENG
Suppose that X n has a distribution belonging to C. By relation (4.9) of Chen [4] (p. 1041), we have where T * n is a positive r.v. independent of X n and distributed by Then, by Lemma 3.3, the distribution of X n e −L θ (T * n ) belongs to C. Hence, H belongs to C. The similar discussion can be done for the regular variation class (R −α ) if we use Lemma 3.4.
In this paper, we obtain an asymptotic estimate for Ψ(x) under the Assumption 1-3 and apply the estimate to the special case that H belongs to R −α . Notice that we do not make any special assumption on the dependence structure of (X, T ). The remaining part of this paper is organized as follows. In Section 2, we introduce some notations and state our main results. In Section 3, we provide some lemmas and prove the main results of the paper.
2. Notations and main results. In this paper, C represents a positive constant without relation to x and may vary from place to place. Hereafter, all limit relations are for x → ∞ unless stated otherwise. For two positive functions a(·) and b(·), In order to facilitate subsequent expression, we denote

Now we recall several classes of heavy-tailed distributions. A distribution F belongs to the dominated variation class (denoted by
< ∞ for any 0 < y < 1. A distribution F belongs to the regular variation class (denoted by R −α ) if F (x) > 0 for all x ∈ R and lim x→∞ F (xy) F (x) = y −α for some α > 0 and all y > 0.
It is well known that R −α ⊂ C ⊂ D ∩ L.
Besides that, the upper Matuszewska index J + F and lower Matuszewska index Theorem 2.2. Replacing "H ∈ C" and "0 < J − H ≤ J + H < κ θ " with "H ∈ R −α for some 0 < α < κ θ " among the conditions of Theorem 2.1, we have 3. Proofs of the main results. By convention, an empty sum is 0 and an empty product is 1.
3.1. Some lemmas. By Proposition 2.2.1 in Bingham et al. [2], for a distribution F ∈ D and arbitrarily fixed p > J + F , there exist positive constants C p and D p such that holds for all x ≥ y ≥ D p . Fixing the variable y leads to The following fundamental lemmas will be used.
Lemma 3.1. Let X and Y be two independent and nonnegative random variables, where X is distributed by F . If F ∈ D, then for arbitrarily fixed δ > 0 and p > J + F , there exists a positive constant C without relation to δ and Y such that for all large x, Proof. See Lemma 3.2 in Heyde and Wang [11].
Lemma 3.2. Let X and Y be two independent and nonnegative random variables, where X is distributed by F . If F ∈ D with J − F > 0, then, for any fixed δ > 0 and 0 < p 1 < J − F ≤ J + F < p 2 < ∞, there exists a positive constant C without relation to δ and Y , such that for all large x, Proof. See Lemma 3 in Guo and Wang [10]. Lemma 3.3. Let X and Y be two independent random variables, X be real-valued and distributed by F and Y be nonnegative and nondegenerate at 0. If F ∈ C and EY p < ∞ for some p > J + F , then the distribution of XY belongs to C and P(XY > x) F (x).
Proof. See Lemma 2.4 and Lemma 2.5 in Wang et al. [18]. Lemma 3.4. Let X and Y be two independent random variables, X be real-valued and distributed by F and Y be nonnegative and nondegenerate at 0. If F ∈ R −α for some 0 < α < ∞ and EY p < ∞ for some p > α. Then, the distribution of XY belongs to R −α and P(XY > x) ∼ F (x)EY α .
Proof. The complete proof can be found in Breiman [3] or Cline and Samorodnitsky [5].
The following lemmas will be used in the proof of Theorem 2.1.
Lemma 3.6. Under the conditions of Theorem 2.1, there exists k * ∈ N + such that holds for any fixed k ≥ k * and 0 < δ < 1.
Proof. By Lemma 3.2 and (17), there exists k * such that for any fixed k ≥ k * , o(H(x)).