Issue 
Wuhan Univ. J. Nat. Sci.
Volume 29, Number 1, February 2024



Page(s)  21  28  
DOI  https://doi.org/10.1051/wujns/2024291021  
Published online  15 March 2024 
Mathematics
CLC number: O211
Uniform Asymptotics for FiniteTime Ruin Probabilities of Risk Models with NonStationary Arrivals and Strongly Subexponential Claim Sizes
^{1}
School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou 215009, Jiangsu, China
^{2}
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, Sichuan, China
^{†} Corresponding author. Email: beewky@vip.163.com
Received:
22
April
2023
This paper considers the one and twodimensional risk models with a nonstationary claimnumber process. Under the assumption that the claimnumber process satisfies the large deviations principle, the uniform asymptotics for the finitetime ruin probability of a onedimensional risk model are obtained for the strongly subexponential claim sizes. Further, as an application of the result of onedimensional risk model, we derive the uniform asymptotics for a kind of finitetime ruin probability in a two dimensional risk model sharing a common claimnumber process which satisfies the large deviations principle.
Key words: onedimensional risk model / twodimensional risk model / large deviations principle / finitetime ruin probability / heavytailed distributions
Cite this article: XU Chenghao, WANG Kaiyong, PENG Jiangyan. Uniform Asymptotics for FiniteTime Ruin Probabilities of Risk Models with NonStationary Arrivals and Strongly Subexponential Claim Sizes[J]. Wuhan Univ J of Nat Sci, 2024, 29(1): 2128.
Biography: XU Chenghao, male, Master candidate, research direction: actuarial science. Email: 534759246@qq.com
Fundation item: Supported by the 333 High Level Talent Training Project of Jiangsu Province, the National Natural Science Foundation of China (71871046) and Science and Technology Projects of Sichuan Province (2021YFQ0007)
© Wuhan University 2023
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 onedimensional risk model, in which the surplus at time is described as
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 claimarrival times, which constitute the claimnumber process
with a finite mean function , , where and by convention. The nonnegative random variables are the claim interarrival times, which are independent of . For the risk model (1), the finitetime ruin probability up to time is defined as
The risk model (1) has been widely studied and results under various conditions are presented. For the uniform asymptotics of the finitetime ruin probability as , when are i.i.d., Tang ^{[1]} investigated the case that the claim sizes have consistentlyvaryingtailed 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 twodimensional 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 interarrival times are i.i.d or have some dependence structures. Few articles have studied the claimnumber process is nonstationary. In fact, a nonstationary claimnumber 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 lighttailed claim sizes. Recently, Refs.[10,11] considered the claimnumber 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 semicontinuous 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 claimnumber process satisfying the LDP and investigates the uniform asymptotics of the finitetime 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 twodimensional risk model and investigates a kind of finitetime 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 heavytailed distributions. Some subclasses of heavytailed distribution class will be given. Say that a distribution on is heavytailed if for any ,
One of the important distribution classes of heavytailed distributions is the consistentlyvaryingtailed 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 longtailed 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 wellknown 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 claimnumber 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
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 ,
and
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
and
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 ,
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 ,
So, for all and sufficiently large ,
By (10), it holds for all and sufficiently large that
which shows that
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
Combining with (12) yields that
By (11) and (13), we get
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 ,
It holds for all and that
Since , by Kesten's bound, for , there exists such that for any and ,
Therefore, by and (15)(17),
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 ,
Thus, by (19), for any ,
For the above , let then and as . Thus, for the above and , and for all , by Lemma 1,
where in the last step, we have used and the inequality
where are some constants and is a nonincreasing function on
By (4), , there exists such that and for sufficiently large ,
Since
which combining with (20) and (22) yields that
By (21) and (23), letting and , it holds that
This completes the proof of (6).
2 TwoDimensional Risk Model
In this section, we will apply Theorem 1 to deal with a twodimensional risk model and derive the asymptotics of the finitetime ruin probability of a twodimensional risk model.
2.1 Risk Model
In recent years, more and more scholars begin to study different twodimensional risk models. In this section, we consider the following twodimensional risk model in which the surplus at time is described as
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 claimarrival times, which constitute the claimnumber process .
The claim interarrival times are independent of . For the risk model (24), some kinds of finitetime ruin probabilities up to time are defined as
where and
where .
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 interarrival 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 claimnumber process satisfying the LDP (i.e. Assumption A). They obtained the uniform asymptotics of the finitetime ruin probability for the claim sizes belonging to the class . In the following we still consider the risk model (24) with a claimnumber process , which satisfies the LDP and investigate the uniform asymptotics of the finitetime 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 twodimensional risk model (24). Suppose that Assumption A holds. If
and
then
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
Proof Since , by Corollary 3.16 of Foss et al^{[20] }we know that (27) holds. Hence, by (27) for sufficiently large ,
holds uniformly for .
According to the definition of , we get that and . So is longtailed. 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
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 nonstationary claimnumber process and obtain the uniform asymptotics for the finitetime ruin probability of a onedimensional risk model for the strongly subexponential claim sizes when the claimnumber process satisfies the large deviations principle. Further, applying Theorem 1 the uniform asymptotics for a kind of finitetime ruin probability in a twodimensional risk model have been presented.
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