Uniform Asymptotics for Finite-Time Ruin Probabilities of Risk Models with Non-Stationary Arrivals and Strongly Subexponential Claim Sizes

: This paper considers the one-and two-dimensional risk models with a non-stationary claim-number process. Under the assump‐ tion that the claim-number process satisfies the large deviations principle, the uniform asymptotics for the finite-time ruin probability of a one-dimensional risk model are obtained for the strongly subexponential claim sizes. Further, as an application of the result of one-dimensional risk model, we derive the uniform asymptotics for a kind of finite-time ruin probability in a two dimensional risk model sharing a common claim-number process which satisfies the large deviations principle.


Introduction
In this section we consider a one-dimensional risk model, in which the surplus at time t ≥ 0 is described as where x ≥ 0 is the initial surplus, c > 0 is the constant premium rate and the claim size {X i i ≥ 1} are independent, identically distributed (i.i. d.) and nonnegative random variables with common distribution F and finite mean.
{τ i i ≥ 1} are the claim-arrival times, which constitute the claim-number process N(t) = sup{i ≥ 0:τ i ≤ t}t ≥ 0 with a finite mean function λ(t) = E(N(t)), t ≥ 0, where sup AE = 0 and τ 0 = 0 by convention.The nonnegative random variables {θ i = τ i -τ i -1 i ≥ 1} are the claim interarrival times, which are independent of {X i i ≥ 1}.For the risk model (1), the finite-time ruin probability up to time t ≥ 0 is defined as The risk model (1) has been widely studied and results under various conditions are presented.For the uniform asymptotics of the finite-time ruin probability ψ(xt) as x ® ¥, when {θ i i ≥ 1} are i.i.d., Tang [1] investi- gated the case that the claim sizes have consistentlyvarying-tailed distributions and obtained the asymptotics of ψ(xt) holds uniformly for t Î Λ ={t ≥ 0: λ(t)> 0}.In the case where the distributions of the claim sizes are from a subclass of subexponential distribution class, Leipus and Šiaulys [2] presented the asymptotics of ψ (xt) holds uniformly for t Î[ f (x)γx], where f (x) is an infinitely increasing function and γ > 0 is a constant.Leipus and Šiaulys [3] and Kočetova et al [4] considered the claim sizes have strong subexponential distributions and showed the asymptotics of ψ(xt) holds uniformly for t Î[ f (x)¥).Yang et al [5] and Wang et al [6] improved the above results by considering the dependent {θ i i ≥ 1}.
Chen et al [7] established a two-dimensional risk model for (1) and obtained some corresponding results for i.i.d.{θ i i ≥ 1}.Chen et al [8] extended the results of Chen et al [7] by considering the dependent {θ i i ≥ 1}.
In the above literatures, they mainly considered the claim inter-arrival times {θ i i ≥ 1} are i.i.d or have some dependence structures.Few articles have studied the claim-number process is non-stationary.In fact, a nonstationary claim-number process may be more practical.Stabile and Torrisi [9] derived the infinite and finite time ruin probabilities for the risk model with a nonstationary Hawkes process and light-tailed claim sizes.Recently, Refs.[10,11] considered the claim-number processes may not be stationary and ergodic and satisfy the large deviations principle (LDP for short).A family of probability measures {μ t } t Î(0¥) on a Hausdorff topological space (MF M ) satisfies the LDP with rate function I: M ®[0¥), if I is a lower semi-continuous function and the following inequalities hold for every Borel set B: ˉI (x) where B o and -B denote the interior and closure of B, respectively, see, e. g., Dembo et al [12] and Bordenave et al [13] .
This section still considers the claim-number process {N(t)t ≥ 0} satisfying the LDP and investigates the uniform asymptotics of the finite-time ruin probability ψ(xt) for the risk model (1).Section 1 presents the main results after introducing necessary preliminaries and the proofs of the main results are given.Section 2 studies a two-dimensional risk model and investigates a kind of finite-time ruin probability by using the results of Section 1.

Preliminaries and Main Results
Hereafter, all limit relationships hold as x ® ¥ unless stated otherwise.For two positive functions a(x) and b(x), we write a(x) ≲ b(x), if lim sup a(x)/b(x)≤ 1; write a(x) ≳ b(x), if lim inf a(x)/b(x)≥ 1 and write a(x)~b(x), if lim a(x)/b(x) = 1.For two positive functions a(xt) and b(xt), we say that a(xt) ≲ b(xt) holds uniformly for and say that a(xt)~b(xt) holds uniformly for t Î D ¹ AE, if a(xt) ≳ b(xt) and a(xt) ≲ b(xt) hold uniformly for t Î D ¹ AE. 1 A is the indicator function of a set A.
In this paper, we will consider the claim sizes have heavy-tailed distributions.Some subclasses of heavytailed distribution class will be given.Say that a distribution V on (-¥¥) is heavy-tailed if for any λ > 0, One of the important distribution classes of heavytailed distributions is the consistently-varying-tailed distribution class C .By definition, a distribution V on (-¥¥) belongs to the class C , denoted by or equivalently, lim A related distribution class is the dominated varying tailed distribution class D .Say that a distribution V on (-¥¥) belongs to the class D , denoted by V Î D , if for any fixed 0 < y < 1, lim sup A distribution V on (-¥¥) is said to be in the long-tailed distribution class L, if for any fixed y > 0, lim An important subclass of the class L is the subexponential distribution class S. By definition, a distribution V on [0¥) is said to be subexponential if where V*V denotes the 2-fold convolution of V.In the case that a distribution V is on (-¥¥), we say that V Î S if the distribution V (x)1 {x ≥ 0} belongs to the class S. It is well-known that these distribution classes have the following inclusions see, e.g., Embrechts et al [14] .Korshunov [15] introduced another subclass of the subexponential distribution class, which is the strongly subexponential distribution class S * .Say that a distribution V on (-¥¥) belongs to the uniformly for u Î[1¥).Korshunov [15] pointed out that the Pareto distribution with parameter exceeding one, the lognormal distribution and the Weibull distribution with suitably chosen parameters belong to the class S * and the class S * almost coincides with the class of subex- ponential distributions with finite means.For the distributions with finite means the following relationships hold see, e.g., Korshunov [15] and Kaas et al [16] .This paper mainly considers the claim-number process {N(t)t ≥ 0} satisfying the LDP.We first present the following assumption.
Assumption A 1) P(N(t)/t Î×) satisfies the LDP with rate function I(×) such that I(x) = 0 if and only if x = z, where z is a positive constant.
As noted in Remark 2.1 of Fu et al [10] , the linear Hawkes process defined in Section 1 of Bordenave et al [13] satisfies Assumption A. One can see Lefevere et al [17] , Macci et al [18] and Jiang et al [19] for some other counting processes satisfying the LDP.
The following is the main result of this section.Theorem 1 Consider the risk model (1).Suppose that Assumption A holds.
holds uniformly for t Î[ f (x)¥), where f: is an infinitely increasing function.
Before giving the proof of Theorem 1, we first present a lemma, which follows from Lemmas 1 and 9 in Korshunov [15] (see also Lemma 2.2 in Leipus and Šiaulys [3] ).
Lemma 1 Let {ξ i i ≥ 1} be i.i.d.random variables with common distribution V and finite mean Eξ 1 < 0.
1) If V Î L , then for sufficiently large x, P ( max holds uniformly for integer n ≥ 1, where ε 1 (x) and ε 2 (x) are some positive vanishing functions as x ® ¥.
In the following we prove Theorem 1.
Proof of Theorem 1 By Assumption A, for any fixed w 1 < z and w 2 > z, there exist some constants δ 1 > 0 and δ 2 > 0 such that I(w 1 ) -δ 1 > 0I(w 2 ) -δ 2 > 0 and for sufficiently large t, P(N(t)/t ≤ w 1 )≤ e -t(I(w 1 ) -δ 1 ) ( and where the facts I(x)> 0 for x ¹ z and I(×) is decreasing on [0z] and increasing on [z¥) have been used.Note that for all x ≥ 0 and t ≥ 0, For any infinitely increasing function f (x), we will prove and hold uniformly for t Î[ f (x)¥), respectively.Firstly, we show the asymptotic upper bound (7).
For any ε > 0x ≥ 0 and t > 0, we have For any δ Î(0v/c), let It follows from the conditions of the risk model that A and B + are independent.
Next we prove the asymptotic lower bound (6).For any ε > 0, by ( 4), for Thus, by (19), for any 0 20) For the above ε > 0, let v͂  c(1/z + ε) -EX 1  then v͂ > 0 and v͂ ® v as ε ® 0. Thus, for the above ε > 0 and M > 0, and for all t Î[ f (x)¥), by Lemma 1, ψ(xt) where in the last step, we have used F Î S * Ì L and the inequality where a ≤ b ≤ c are some constants and g(x) is a nonincreasing function on [ac].

Two-Dimensional Risk Model
In this section, we will apply Theorem 1 to deal with a two-dimensional risk model and derive the asymptotics of the finite-time ruin probability of a twodimensional risk model.

Risk Model
In recent years, more and more scholars begin to study different two-dimensional risk models.In this section, we consider the following two-dimensional risk model in which the surplus at time t ≥ 0 is described as ( ) where x  = (x 1 x 2 ) T is the initial surplus vectors; c = (c 1 c 2 ) T is the vector of constant premium rates; the claim size vectors {( X 1i X 2i ) i ≥ 1} are i.i. d. copies of ( X 1 X 2 ) with nonnegative independent component and marginal distributions F i i = 12, respectively; {τ i i ≥ 1} are the claim-arrival times, which constitute the claimnumber process {N(t)t ≥ 0}.The claim inter-arrival times {θ i = τ i -τ i -1 i ≥ 2θ 1 = τ 1 } are independent of {(X 1i X 2i )i ≥ 1}.For the risk model (24), some kinds of finite-time ruin probabilities up to time t ≥ 0 are defined as ψ max (x t) = P ( t max ≤ t|U i (0) = x i i = 12 )  where t max = inf {s ≥ 0: max{U 1 (s)U 2 (s)}< 0} and ψ sum (x t) = P ( t sum ≤ t|U i (0) = x i i = 12 ) (25) where t sum = inf {s ≥ 0:U 1 (s) + U 2 (s)< 0}.
In some earlier works on the asymptotics of finitetime ruin probabilities, an important assumption is that the two kinds of businesses share a common claimnumber process and the inter-arrival times are independent or have some dependence structure, see, e.g., Li et al [21] , Chen et al [7] , Chen et al [8] , Lu et al [22] and so on.Recently many researchers have paid more attention to some generalizations of risk model (24), such as a risk model with a constant force of interest or stochastic return, see, e. g., Konstantinides et al [23] , Li et al [24] , Li [25] , Yang et al [26] , Cheng and Yu [27] , Cheng et al [28] , Yang et al [29] and so on.
Recently, Fu and Li [10] considered the risk model (24) sharing a common claim-number process satisfying the LDP (i.e.Assumption A).They obtained the uniform asymptotics of the finite-time ruin probability ψ max (x t) for the claim sizes belonging to the class C. In the fol- lowing we still consider the risk model (24) with a claimnumber process {N(t)t ≥ 0}, which satisfies the LDP and investigate the uniform asymptotics of the finite-time ruin probability ψ sum (x t) for the strongly subexponential claim sizes by using Theorem 1.
For the risk model (24), we assume that {X 1i i ≥ 1} { } X 2i i ≥ 1 and {N(t)t ≥ 0} are independent.The follow- ing is the main result of this section.
The proof of the main result will be given in the following subsection.