1. Introduction
In this paper, we consider an insolvency problem for an insurance company that makes investments. The classic insurance risk model has the following relation:
where
denotes the surplus of the insurance company up to time
t,
is the initial reserve,
is the total premiums up to time
t with premium rate
, and
is the total claim sizes up to time
t. Here,
is the sequence of claim sizes whose common inter-arrival times
form a sequence of independent and identically distributed (i.i.d.) random variables. Then the arrival times of the successive claims
, constitute a renewal counting process
with renewal function
, where
is the indicator function of an event
E.
Suppose that the insurance company is allowed to make risk-free or risky assets. From Guo and Wang [
1], we can denote the investment return of the surplus of the insurance company by
, where
can be a stochastic process. Further, the insurance company invests one unit of capital into financial assets at time 0 and receives the benefits of
units at time
t. By (1.1) of Guo and Wang [
1], we can solve the stochastic differential equation satisfied by the surplus process of insurance risk process with investment, and then obtain the integrated risk process
of the insurance company,
This paper considers
with
, where
is a stochastic short-rate process, and the evolution of the interest rate is given by the following Cox–Ingersoll–Ross (CIR) model:
where
m,
l, and
are positive constants,
is a constant, and
is a standard Wiener process.
This paper adopts the following definition of ruin probability which is given by
where
denotes the ruin time with
by convention.
The ruin probability of insurance risk models has been a hot topic of risk theory and actuarial mathematics. The study on the uniform asymptotic estimates for ruin probabilities in insurance risk models with constant force of interest has achieved fruitful results. Some recent works include Chen et al. [
2], Cheng and Yu [
3], and Jiang et al. [
4], etc. For the risk model with investment return process described by geometric Lévy process, there are many studies on the uniform asymptotic analysis for ruin probability. See, for example, Fu and Ng [
5], Guo and Wang [
1], Guo et al. [
6], Li [
7], and Tang et al. [
8], among many others. However, all the aforementioned works did not pay special attention to the case of the uniform asymptotic formula for risk models with risky investments related to the CIR model. In the actuarial literature, many researchers assumed that the claim sizes in one line of business of the insurance company are independent. For example, Fu and Ng [
5], Guo et al. [
6], Jiang et al. [
4], Li [
7], and Tang et al. [
8], etc. However, the assumption of independence between the claim sizes
,
, is too strong. Therefore, some authors started to propose extensions with various dependence structures. Guo and Wang [
1] considered that the claim size sequence
is bivariate upper tail independent. Chen et al. [
2] investigated that the sequences of claim sizes are upper tail asymptotically independent (UTAI). Cheng and Yu [
3] assumed that the sequences of claim sizes are tail asymptotically independent (TAI). Motivated by the references mentioned-above, in this paper we consider that the sequence of claim sizes
possesses an arbitrary dependent structure, and then we derive the uniform asymptotic formula of
for model (
2).
In the rest of this paper,
Section 2 presents the main result after recalling some necessary preliminaries,
Section 3 performs some simulations,
Section 4 establishes some crucial lemmas,
Section 5 proves the main result, and
Section 6 restates the paper’s context and discusses future work.
2. Preliminaries and Main Result
Hereafter, C always stands for a positive constant and may vary in different places. For two positive functions and satisfying we say that holds uniformly for if ; holds uniformly for if ; holds uniformly for if ; and holds uniformly for if . For two real numbers a and b, we write and .
Now recall the definition and some properties of distributions with regularly varying tails. A distribution
F on
belongs to the class of distributions with regularly-varying tails if
for all
and
for some
, denoted by
. By Theorem 1.5.2 of Bingham et al. [
9], the convergence in (
5) is uniform over
for every fixed
; namely,
For a distribution
with some
, according to Bingham et al. [
9] (Proposition 2.2.1) we know that, for any
, there exist two positive constants
and
such that
holds uniformly for all
. From the relation (
7), it follows that
In addition, suppose that a nonnegative random variable
X with distribution function
for some
and a nonnegative random variable
Y are independent. Then, by Lemma 3.2 of Heyde and Wang [
10], there exists some constant
without relation to
Y and
such that, for arbitrarily fixed
and
,
For more details on the distribution class of regular variation, we refer to Bingham et al. [
9] and Embrechts et al. [
11].
For
and the relation (
3), according to the relation (3.30) of Guo and Wang [
1], we obtain
where
Denote by
the set of all
t for which
. With
, it is clear that
For notational convenience, we write for every fixed . As usual, assume that , and are independent. We are now ready to state the main result of this paper:
Theorem 1. Consider the insurance risk model (2) in which the claim sizes form a sequence of identically distributed but not necessarily independent random variables with common distribution for some . Furthermore, we allow arbitrary dependence structures between the claim sizes. If there exists constant such that . Then it holds uniformly for all that An insurance company has to hold enough risk capital so that the ruin probability is sufficiently low. Furthermore, the pricing of related insurance products may use the ruin probability as a trigger. Both cases require an assessment of the ruin probability, making the relation (
11) plays an important role in guidance to insurers and regulators for risk capital calculation.
From Theorem 1, there exists
such that
, hence we have by (
10) that, for
and large
,
. Consequently,
Fix
for
in (
10), then the following relation holds for
:
Fix
in (
10), then we obtain
3. Numerical Simulations
In this section, we illustrate the accuracy of the relation (
11) in Theorem 1. All the numerical simulations are carried out on R software.
For simplicity, we assume that the claim inter-arrival times are independent and identically distributed exponential random variables with parameter
. Moreover, the successive claims form a sequence of i.i.d. random variables with the generic random variable
X. Let
X follow a Pareto distribution
F with shape parameter
and scale parameter
, which means
,
. Then, the asymptotic estimation of ruin probability
can be rewritten as
The parameters in this subsection are given as: , , , , , , , , , and the initial capital , 400, 600, 800, 1000.
We employ to represent the Monte Carlo (MC) simulation result of the ruin probability . The procedure of simulation of is the following:
- 1.
Assign a value for variable x and set , , and ;
- 2.
Generate random variable X with X following Pareto distribution F and following exponential distribution with parameter , and then set ;
- 3.
Set
. If
, set
. If
, divide the interval
into 30 pieces, and denote these points as
,
, ⋯,
. According to Glasserman [
12] (p. 124), we can simulate
, ⋯,
, and then we can simulate
, ⋯,
from
,
. Calculate
If , then and . If not, set , and repeat Steps 2 and 3;
- 4.
Set and . Repeat Steps 2 and 3 until ;
- 5.
Calculate .
When the initial capital
, 400, 600, 800, 1000, the comparison of asymptotic estimate and MC simulation of
can be seen in
Table 1. From
Table 1, we observe that the ruin probability decreases as
x becomes large, and the ratio
becomes closer to 1 as
x becomes large.