Open Access
 Issue Wuhan Univ. J. Nat. Sci. Volume 29, Number 1, February 2024 21 - 28 https://doi.org/10.1051/wujns/2024291021 15 March 2024

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## 0 Introduction

In this section we consider a one-dimensional risk model, in which the surplus at time is described as

(1)

where is the initial surplus, is the constant premium rate and the claim size are independent, identically distributed (i.i.d.) and nonnegative random variables with common distribution and finite mean. are the claim-arrival times, which constitute the claim-number process

with a finite mean function , , where and by convention. The nonnegative random variables are the claim inter-arrival times, which are independent of . For the risk model (1), the finite-time ruin probability up to time is defined as

(2)

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 as , when are i.i.d., Tang [1] investigated the case that the claim sizes have consistently-varying-tailed distributions and obtained the asymptotics of holds uniformly for . 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 holds uniformly for , where is an infinitely increasing function and 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 holds uniformly for . Yang et al[5] and Wang et al[6] improved the above results by considering the dependent . Chen et al [7] established a two-dimensional risk model for (1) and obtained some corresponding results for i.i.d. . Chen et al [8] extended the results of Chen et al [7] by considering the dependent .

In the above literatures, they mainly considered the claim inter-arrival times are i.i.d or have some dependence structures. Few articles have studied the claim-number process is non-stationary. In fact, a non-stationary 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 non-stationary 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 on a Hausdorff topological space satisfies the LDP with rate function , if is a lower semi-continuous function and the following inequalities hold for every Borel set :

where and denote the interior and closure of , respectively, see, e.g., Dembo et al [12] and Bordenave et al[13].

This section still considers the claim-number process satisfying the LDP and investigates the uniform asymptotics of the finite-time ruin probability 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.

## 1 Preliminaries and Main Results

Hereafter, all limit relationships hold as unless stated otherwise. For two positive functions and , we write , if ; write , if and write , if . For two positive functions and , we say that holds uniformly for , If

;

say that holds uniformly for , if

;

and say that holds uniformly for , if and hold uniformly for . is the indicator function of a set .

In this paper, we will consider the claim sizes have heavy-tailed distributions. Some subclasses of heavy-tailed distribution class will be given. Say that a distribution on is heavy-tailed if for any ,

One of the important distribution classes of heavy-tailed distributions is the consistently-varying-tailed distribution class . By definition, a distribution on belongs to the class , denoted by , if

or equivalently,

A related distribution class is the dominated varying tailed distribution class . Say that a distribution on belongs to the class , denoted by , if for any fixed ,

A distribution on is said to be in the long-tailed distribution class , if for any fixed ,

An important subclass of the class is the subexponential distribution class . By definition, a distribution on is said to be subexponential if

where denotes the -fold convolution of . In the case that a distribution is on , we say that if the distribution belongs to the class . 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 . Say that a distribution on belongs to the class , if and the distribution defined by

satisfies

uniformly for . 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 and the class almost coincides with the class of subexponential 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 satisfying the LDP. We first present the following assumption.

Assumption A 1) satisfies the LDP with rate function such that if and only if , where is a positive constant.

2) is increasing on and decreasing on .

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. If and , then

(3)

holds uniformly for , where 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 be i.i.d. random variables with common distribution and finite mean .

1) If , then for sufficiently large ,

holds uniformly for integers ;

2) If , then for sufficiently large ,

holds uniformly for integer , where and are some positive vanishing functions as .

In the following we prove Theorem 1.

Proof of Theorem 1   By Assumption A, for any fixed and , there exist some constants and such that and for sufficiently large ,

(4)

and

(5)

where the facts for and is decreasing on and increasing on have been used.

Note that for all and ,

For any infinitely increasing function , we will prove

(6)

and

(7)

hold uniformly for , respectively.

Firstly, we show the asymptotic upper bound (7). For any and , we have

For any , let

It follows from the conditions of the risk model that and are independent.

We first estimate . Let . Therefore, for all and

Using the line of the proof of Proposition 2.1 of Leipus and Šiaulys [3], we know that for all and ,

where is a positive function satisfying Let denote the integral of the right side in the above inequality and be the distribution of the random variable . By Fubini's theorem, for all and ,

Let in (5). Since , it knows that . By (5) there exists a constant such that for sufficiently large ,

(8)

where

and

Thus for

Since , it holds that By Corollary 3.18 of Foss et al[20],

Consequently, there exists a positive function such that for sufficiently large ,

(9)

So, for all and sufficiently large ,

(10)

By (10), it holds for all and sufficiently large that

which shows that

(11)

In the following, we deal with . Using (8), for sufficiently large , we have

Since and as , for sufficiently large , it holds that . Therefore, by , for all and , it holds that

(12)

Combining with (12) yields that

(13)

By (11) and (13), we get

(14)

Next, we will estimate the asymptotic upper bound of . We first deal with the process . By (5), for , there exists such that and for all and sufficiently large ,

(15)

It holds for all and that

(16)

Since , by Kesten's bound, for , there exists such that for any and ,

(17)

Therefore, by and (15)-(17),

(18)

Thus, by (14) and (18),

This completes the proof of (7).

Next we prove the asymptotic lower bound (6). For any , by (4), for , there exists such that and for sufficiently large ,

(19)

Thus, by (19), for any ,

(20)

For the above , let then and as . Thus, for the above and , and for all , by Lemma 1,

(21)

where in the last step, we have used and the inequality

where are some constants and is a non-increasing function on

By (4), , there exists such that and for sufficiently large ,

(22)

Since

which combining with (20) and (22) yields that

(23)

By (21) and (23), letting and , it holds that

This completes the proof of (6).

## 2 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 two-dimensional risk model.

### 2.1 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 is described as

(24)

where is the initial surplus vectors; is the vector of constant premium rates; the claim size vectors are i.i.d. copies of with nonnegative independent component and marginal distributions , respectively; are the claim-arrival times, which constitute the claim-number process .

The claim inter-arrival times are independent of . For the risk model (24), some kinds of finite-time ruin probabilities up to time are defined as

where and

(25)

where .

In some earlier works on the asymptotics of finite-time ruin probabilities, an important assumption is that the two kinds of businesses share a common claim-number 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 for the claim sizes belonging to the class . In the following we still consider the risk model (24) with a claim-number process , which satisfies the LDP and investigate the uniform asymptotics of the finite-time ruin probability for the strongly subexponential claim sizes by using Theorem 1.

For the risk model (24), we assume that and are independent. The following is the main result of this section.

Theorem 2   Consider the two-dimensional risk model (24). Suppose that Assumption A holds. If

and

then

(26)

holds uniformly for as , where is an infinitely increasing function.

The proof of the main result will be given in the following subsection.

### 2.2 Proof of Theorem 2

The following lemma is crucial to prove Theorem 2.

Lemma 2   Let and be nonnegative random variables with distributions and , respectively. If and , then and

(27)

Proof   Since , by Corollary 3.16 of Foss et al[20] we know that (27) holds. Hence, by (27) for sufficiently large ,

(28)

holds uniformly for .

According to the definition of , we get that and . So is long-tailed. It follows from that . Again using Corollary 3.16 of Foss et al[20], by (28) we have and

Thus, by Corollary 3.13 of Foss et al [20], which means .

Proof of Theorem 2   Note that

Since and , by Lemma 2 we get and

(29)

Thus, by Theorem 1 and (29) for sufficiently large , it holds uniformly for that

This completes the proof of Theorem 2.

## 3 Conclusion

We consider the risk models with a non-stationary claim-number process and obtain the uniform asymptotics for the finite-time ruin probability of a one-dimensional risk model for the strongly subexponential claim sizes when the claim-number process satisfies the large deviations principle. Further, applying Theorem 1 the uniform asymptotics for a kind of finite-time ruin probability in a two-dimensional risk model have been presented.

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