1. Introduction
In traditional studies, many researchers have investigated ruin probability problems of insurers under unidimensional models. For example, ref. [
1] studied ruin probability problems with constant interest force. Other studies about these problems can be found in [
2,
3,
4,
5]. An assumption behind these models is that the insured businesses homogeneous and can be described by a unidimensional model; however, this assumption is too strong. Thus, bidimensional or multidimensional insurance risk models have received growing interest in recent years, such as [
6,
7,
8]. Various assumptions have been considered regarding the claim arrival process and the distribution of claim amounts; see, e.g., [
9,
10,
11,
12]. Ref. [
13] considered finite-time ruin probabilities for nonstandard bidimensional renewal risk models with constant interest forces and diffusion generated by Brownian motions; they assumed that the two Brownian motions
and
are mutually independent. Similar results were obtained by [
14], although they considered dependent subexponential claims. More papers can be found in [
15,
16], and the references therein. In this paper, we consider uniform asymptotics for the finite-time ruin probabilities for several bidimensional risks models with constant interest force and correlated Brownian motions, meaning that the businesses of the insurer have a relationship with each other. We introduce risk models and different types of ruin times with corresponding ruin probabilities as follows.
The bidimensional risk model
is the surplus vector of an insurance company at time
; in this paper, we state this formally as
where
stands for the initial surplus vector and
for the total premiums received up to time
t; here,
,
are mutually independent. Moreover,
stands for the interest rate and
for the total amount of claims vector up to time
t. Here,
denote pairs of claims with arrival times that constitute a counting process vector
, where
, while
,
are mutually independent. The process
is a Poisson process with intensity
, and
is a sequence of independent copies of the random pair
with the joint distribution function
and the marginal distribution functions
and
. For all vectors, the
s and
consist of only non-negative components
. Moreover, each
is a non-decreasing and right-continuous stochastic process. The vector
denotes a standard bidimensional Brownian motion with a constant correlation coefficient
, while
and
are constants. For simplicity, we assume that
,
and
are independent and that both of them are independent of
. To avoid the certainty of ruin in each class, we assume that the following safety loading conditions hold when
:
In this paper, we consider the following four types of ruin probabilities. For a finite horizon
, we define
where
where
and
where
where
and
with
by convention.
We remark that the probability in (2) denotes the probability of ruin occurring when both
and
are below zero at the same time within finite time
, the probability in (3) denotes the probability of ruin occurring when at least one of
is below zero within finite time
, the probability in (4) denotes the probability of ruin occurring when the total of
and
is negativ within finite time
, and the probability in (5) denotes the probability of ruin occurring when both
and
are below zero, not necessarily simultaneously, within a finite time
.
represents a more critical time than
, and the ruin probability defined by
is reduced to that in the unidimensional model. The following relation between the four ruin probabilities defined above holds:
and
The rest of this paper is organized as follows. In
Section 2 we review the related results after briefly introducing preliminaries about heavy-tailed distributions, in
Section 3 we provide several important definitions and lemmas, and the main results and the proof procedure are presented in
Section 4.
2. Review of Related Results
Unless otherwise stated herein, all limit relations are for . We denote and if and , respectively, and if both, where, and are two positive functions. Let be the convolution of the distributions and let denote the n-fold convolution of a distribution F.
In this section, we review definitions and properties that are relevant to the results of this paper, considering only the case of the distribution of heavy-tail claims. An r.v. X or its d.f. satisfying for all is called heavy-tailed to the right, or simply heavy-tailed, if for all . In the following, we recall several important classes of heavy-tailed distributions.
F is a long tailed distribution, written as
, if
holds for some
. Note that the convergence is uniform over
t in compact intervals. If
holds
, then
F is a subexponential distribution on
, written as
. For some
, if
holds,
F is said to have a dominatedly varying tailed distribution, written as
. We call
F a consistently varying tailed distribution, written as
, if
holds. A distribution
F is extended regularly-varying tailed, written as
for some
, if
holds for
.
It is obvious that the following formula holds:
There are many other references to heavy-tailed distributions; readers may refer to [
17,
18,
19,
20,
21,
22] among others.
The asymptotic behavior of the finite-time ruin probability of bidimensional or multidimensional risk models has previously been investigated by [
23]. They proved that under the conditions
,
, and
, it is the case that
and the claim vector
consist of independent components
Under the conditions
,
, and
, it is the case that
are deterministic linear functions, and both the claim vector
and the bidimensional Brownian motion
consist of independent components. Li et al. [
12] found that for each fixed time
,
Chen et al. [
11] investigated the uniform asymptotics of
and
for an ordinary renewal risk model with the claim amounts belonging to the consistently varying tailed distributions class for large
T. Zhang and Wang [
24] considered model (1) with
and assumed that all sources of randomness,
,
,
,
and
are mutually independent. They obtained that if
for some
, then, for each fixed time
,
The analogous result for multidimensional risk models can be found in Asmussen and Albrecher [
17].
3. Some Lemmas
Before providing the main results, we first provide several lemmas.
Lemma 1. If , then for each there exists some constant such that the inequalityholds for all and . Proof. See Lemma 1.3.5 of Embrechts et al. [
25]. □
Lemma 2. Let and be two distribution functions. If and , then we have as .
Proof. See Proposition 1 of Embrechts et al. [
25]. □
Lemma 3. Consider a unidimensional risk model If , then the ruin probability with finite-horizon T satisfies Proof. Clearly, on the one hand,
where we have used the fact that
and the dominated convergence theorem.
On the other hand,
where we have used Lemma 2 and the fact that
Per Lemma 1 and dominated convergence theorem, we have
The result follows from (8) and (9). □
Lemma 4. Consider a unidimensional risk model If , then the ruin probability with finite-horizon T satisfies Proof. By simply modifying the proof of Lemma 3, we have
where in the last step we use (28) from [
26]. Here,
are the arrival times of the Poisson process
. In fact,
and we have that
Upon a trivial substitution, the required result is implied. □
Definition 1. - (i)
Two processes and are said to be positively associated iffor all non-decreasing real valued functions f and g such that covariance exists, all , and all . - (ii)
Two processes and are said to be negatively associated iffor all non-decreasing real valued functions f and g such that covariance exists, all, and all
.
Definition 2. Two processes and are said to be positively (negatively) quadrant-dependent iffor all and for all . It is well known (cf. Ebrahimi [27]) that being positively (negatively) associated implies that and are positively (negatively) quadrant-dependent. Let be a standard bidimensional Brownian motion with constant correlation coefficient . For notional convenience, for we write It is well known that for . The following lemma is essential to proving our main results. Moreover, it is of independent interest.
Lemma 5. For any , if , thenandIf , thenand Proof. For any
, we have
. It follows from the Theorem in Pitt [
28] that
is necessary and sufficient for
to be positively associated, as
is bivariate normal, which implies that
is positively quadrant-dependent. Thus, (11) holds. To prove (12), we use (11) and the facts that
and
is a standard bidimensional Brownian motion with correlation coefficient
. Inequalities (13) and (14) can be proved similarly. This completes the proof.
For
, consider a bidimensional Gaussian process
, where
is a standard bidimensional Brownian motion with constant correlation coefficient
. For
, we can write
The following lemma is an extension of Lemma 5. □
Lemma 6. For any , if , thenandIf , thenand Remark 1. Several distributions of interest are available in closed form (see, e.g., He, Keirstead, and Rebholz [29]). These include the joint distributions of , , , and so on. However, those closed-form results cannot apply our proofs to the main results. The results of Lemmas 5 and 6 cannot be obtained from the results of Shao and Wang [30]. Lemma 7. Let be a Poisson process with arrival times . Considering for arbitrarily fixed and , the random vector is equal in distribution to the random vector , where denote the order statistics of n i.i.d. (0, 1) uniformly distributed random variables .
Proof. See Theorem 2.3.1 of Ross [
26]. □
Lemma 8. Let X and Y be two independent and non-negative random variables. If X is subexponentially distributed while Y is bounded and non-degenerate at 0, then the product is subexponentially distributed.
Proof. See Corollary 2.3 of Cline and Samorodnitsky [
19]. □
The following result is due to Tang [
1].
Lemma 9. Let X and Y be two independent random variables with distributions and . Moreover, let Y be non-negative and non-degenerate at 0. Then, 4. Main Results and Proofs
In this paper, we establish new results for the finite-time ruin probabilities. Unlike the above-motioned articles, we assume that the two Brownian motions and are correlated with a constant correlation coefficient . The following are the main results of this paper.
Theorem 1. Consider the insurance risk model introduced in Section 1. Assume that , , and that , , , , , are mutually independent. - (a)
If , then, for each fixed time , - (b)
If , then, for each fixed time ,
Proof. First, we establish the asymptotic upper bound for
. Clearly,
Because
, by using (14) we have
Using the independence of
and
, we have
Substituting (19) and (20) into (18) and using the dominated convergence theorem, we obtain
where in the second step we have used Lemma 2 and the fact that
Next, we establish the asymptotic lower bound for
. Clearly,
where
can be written as
Here,
and
For large constants
and
, we can further write
as
First, we consider
. Then, per Lemma 9, it holds uniformly for all
that
and it holds uniformly for all
that
Using Lemma 1 and the dominated convergence theorem, we obtain
Thus,
Now, we consider
. Using (25), Lemma 1, and the dominated convergence theorem,
Thus,
Likewise,
Finally, we deal with
:
from which we obtain
From (23) and (27)–(30), we obtain
Now, it follows from (22), (31), and the dominated convergence theorem that
from which, along with (21), we obtain (15).
Note that
from which, along with (18) and (21), we have
Thus, it is the case that
, as
From (6), we have
From Lemma 3, we can obtain (16).
Next, we prove relation (17). Using Theorem 7.2 in Ikeda and Watanabe [
31] (and see Yin and Wen [
32]), for all
we have
where ‘
’ denotes equality in distribution,
W is a standard Brownian motion independent of
,
,
,
, and
. Thus, for all
,
can be written as
Applying Lemma 3 to this model, we find that if
, then
where, in the last step, we have relied on the statement in [
33] (and see Geluk and Tang [
34]) that
This ends the proof of Theorem 1. □
Remark 2. Letting and in Theorem 1, we obtain Theorem 1 in [12]. Theorem 2. Consider the insurance risk model introduced in Section 1. Assume that , , and that , , , , , are mutually independent. - (a)
If , then for each fixed time , - (b)
If , then for each fixed time ,
In particular, if there are two positive constants and such that , , then Proof. We can write
as
For
and each
or 2, we have
It follows that
satisfies
where we have used Lemma 7 in the last steps. Because
, using Lemma 6, we have
Using the independence of
and
, we have
Substituting (37) and (38) into (36) and using
and
uniformly for
, we obtain
We apply Proposition 5.1 of Tang and Tsitsiashvili [
22], which says that for i.i.d. subexponential random variables
and for arbitrarily
a and
b where
, the relation
holds uniformly for
. Hence, by conditioning on
we find that where
by substituting (40) into (39) and using the dominated convergence theorem, we obtain
Next, we establish the asymptotic lower bound for
. Clearly,
where, for some positive constants
c and
d,
Here,
and
Per Lemma 8, we know that
, as all
. Then, invoking Lemma 9, we obtain
uniformly for all
and
, respectively. Now, using the same argument by which we reached (31), we have
Now, it follows from (42), (43), Lemma 1, and the dominated convergence theorem that
or, equivalently,
from which, along with (41), we obtain (32).
The relation (33) follows from (6) and Lemma 4 because, as above,
From (6), we have
From Lemma 4, we have
Thus, we have completed the proof of (33).
Next, we prove relation (34). Similarly, for all
, we have
where
is a standard Brownian motion independent of
,
,
,
, and
.
From Lemma 4, we have
Let
; then,
. Therefore,
This completes the proof of (34). The result (35) follows from (34) and Lemma 3.1 in [
5]. This ends the proof of Theorem 2. □
Remark 3. When letting , , in Theorem 2, we obtain the result in Liu et al. [23]. Theorem 3. Consider the insurance risk model introduced in Section 1. Assume that , and , , , , , and , are mutually independent. (a) If , then for each fixed time , (b) If , where ξ is a random variable independent of and and ; then, for each fixed time , Proof. As the proof is similar to that of Theorem 1, we only provide the main steps. First, we establish the asymptotic upper bound for
. Clearly,
Because
, using (14), we have
Substituting (49) into (48), we obtain
where in the last step we have used Lemma 3.
Next, we establish the asymptotic lower bound for
. Clearly,
where
Using the same arguments as those used to prove (31), we obtain
from which, together with (51), we have
The proof of (46) is straightforward, and is omitted here. Next, we prove (47). Using the properties of two independent compound Poisson processes and two independent Brownian motions, for all
we have
where
is a standard Brownian motion,
is a Poisson process with intensity
, and
is a Bernoulli random variable with
. Moreover,
,
,
,
,
,
,
, and
are independent. Applying Lemma 3 to this model, we obtain
and result (47) follows (c.f. Kaas et al. [
35].)
This ends the proof of Theorem 3. □
Theorem 4. Consider the insurance risk model introduced in Section 1. Assume that , , and that , , , , , , and are mutually independent. (a) If , then for each fixed time , (b) If , where ξ is defined as in Theorem 3, then for each fixed time , Proof. As in the proof of Theorem 2, we have
It follows that
The asymptotic lower bound for can be established similarly.
The relation (53) follows from (6), Lemma 4, and the fact that
Finally, we prove (54). Using the same arguments as above, we have
where
,
,
are the same as in the proof of Theorem 3. It follows from Lemma 4 that
and the result (54) follows, as
This completes the proof of Theorem 4. □