Asymptotic Ruin Probability of a Bidimensional Risk Model Based on Entrance Processes with Constant Interest Rate

Abstract

In this paper, the risk model with constant interest based on an entrance process is investigated. Under the assumptions that the entrance process is a renewal process and the claims sizes satisfy a certain dependence structure, which belong to the different heavy-tailed distribution classes, the finite-time asymptotic estimate of the bidimensional risk model with constant interest force is obtained. Particularly, when inter-arrival times also satisfy a certain dependence structure, these formulas still hold.

1. Introduction

In this paper, we investigate a bidimensional risk model based on entrance processes, in which an insurance company operates two kinds of business. Suppose that the initial insurance fund for i-th class is $x_i$ and $S_j^{(i)}$ is entry time of the j-th policy with $0<S_1^{(i)}<S_2^{(i)}<\cdots$ and $S_j^{(i)}={\sum }_{k=1}^j{\theta }_k^{(i)}$, ${\theta }_1^{(i)}=S_1^{(i)},i=1,2$. The corresponding renewal process, up to the time t, is

$$\begin{array}{c}\hfill N_i(t)=\sum _{j=1}^{\infty }{I}_{\{S_j^{(i)}\le t\}}.\end{array}$$

where $I_{\{·\}}$ is the indicator function.

Denote renewal function by $m_i(t)=E(N_i(t))={\sum }_{j=1}^{\infty }P(S_j^{(i)}\le t),t\ge 0$ and suppose the $m_i(t)<\infty$ for all $0<t<\infty$ and $m_i(0)=0,i=1,2$. Let the validity time of the j-th policy be $\{C_j^{(i)},j=1,2,\cdots \}$ with probability $P(C_j^{(i)}={\alpha }_{\ell })=p_{\ell }$, $\ell =1,2,\cdots ,K$, and ${\alpha }_1<{\alpha }_2<\cdots <{\alpha }_K$, where they are independent and identically distributed. The premium is $f_i(C_j^{(i)})$ and $f_i(·)$ is a strictly increasing function. $D_j^{(i)}$ is claim time of the j-th policy and independent and identically distributed function $H_i(·)$. $Y_j^{(i)}$ is the j-th claim size and identically distributed function $F_i(·)$. Suppose that $D_j^{(i)}$, $Y_j^{(i)}$ and $C_j^{(i)}$ have the same distributions with random variables $D_i$, $Y_i$ and $C_i$, respectively.

Assume that an insurance company invests risk free market with force of interest $\delta >0$, then up to time t, the surplus process of the insurance company is written as:

$$\begin{array}{cc}\left(\begin{array}{c}{R}_1(t)\\ R_2(t)\end{array}\right)\hfill & =\left(\begin{array}{c}{x}_1{e}^{\delta t}\\ x_2{e}^{\delta t}\end{array}\right)+\left(\begin{array}{c}{\sum }_{j=1}^{N_1(t)}{f}_1(C_j^{(1)})e^{\delta (t-S_j^{(1)})}\\ {\sum }_{j=1}^{N_2(t)}{f}_2(C_j^{(2)})e^{\delta (t-S_j^{(2)})}\end{array}\right)\hfill \\ & -\left(\begin{array}{c}{\sum }_{j=1}^{N_1(t)}{Y}_j^{(1)}{e}^{\delta (t-S_j^{(1)}-D_j^{(1)})}{I}_{\{S_j^{(1)}+D_j^{(1)}\le t,D_j^{(1)}\le C_j^{(1)}\}}\\ {\sum }_{j=1}^{N_2(t)}{Y}_j^{(2)}{e}^{\delta (t-S_j^{(2)}-D_j^{(2)})}{I}_{\{S_j^{(2)}+D_j^{(2)}\le t,D_j^{(2)}\le C_j^{(2)}\}}\end{array}\right).\hfill \end{array}$$

(1)

Now the following two types of ruin times for a bidimensional risk model based on entrance processes are considered. we define the first time when both $R_1(t)$ and $R_2(t)$ become negative by

$$\begin{array}{c}\hfill {\tau }_{\max }(x_1,x_2)=\inf \{t:\max \{R_1(t),R_2(t)\}<0|R_i(0)=x_i,i=1,2\},\end{array}$$

the first time when both $R_1(t)$ or $R_2(t)$ become negative by

$$\begin{array}{c}\hfill {\tau }_{\min }(x_1,x_2)=\inf \{t:\min \{R_1(t),R_2(t)\}<0|R_i(0)=x_i,i=1,2\}.\end{array}$$

Then we define the corresponding ruin probabilities with the finite time $t>0$ respectively by

$$\begin{array}{c}\hfill {\psi }_{\max }(x_1,x_2,T)=P({\tau }_{\max }(x_1,x_2)\le T)=P(\bigcap _{i=1}^2\{R_i(s)<0\}\mathrm{for}\mathrm{some}0\le s\le T),\end{array}$$

(2)

and

$$\begin{array}{c}\hfill {\psi }_{\min }(x_1,x_2,T)=P({\tau }_{\min }(x_1,x_2)\le T)=P(\bigcup _{i=1}^2\{R_i(s)<0\}\mathrm{for}\mathrm{some}0\le s\le T).\end{array}$$

(3)

We know that Li et al. (2005) put forward into a new model ($\mathrm{LIG}$ $\mathrm{model}$) based on an entrance process and discussed asymptotic normality of the risk process. Furthermore, Some scholars got some conclusions through the study of the $\mathrm{LIG}$ model. Li and Kong (2007) discussed the weak convergence properties of the model. Xiao et al. (2008) studied some limit properties of the model under constant interest. Xiao and Tang (2009) studied the infinite ruin probability with constant interest within Poisson process and class ${\mathcal{R}}_{-\alpha }$. Xiao et al. (2013) discussed the ruin probability of $\mathrm{LIG}$ model. It is clear that the above literatures are improved and investigated for one-dimensional risk model based on entrance processes. Recently, people have been interested in two-dimensional risk model, see, for example, Chan et al. (2003); Li et al. (2007); Zhang and Wang (2012) and so on. It is well known that these literatures are investigated in the classical model, and the risk model based on entrance processes is more important and actual. Therefore, on the basis of above literatures, we consider a bidimensional risk model based on entrance processes.

Because Theorem 1 of Xiao and Tang (2009) is obtained under the Poisson process and the regular variation class, we know that this is far from the actual. Hence, in this paper, we consider the $\mathrm{LIG}$ model and obtain the finite-time ruin probability under the class $\mathcal{L}\cap \mathcal{D}$ with constant interest when claim sizes satisfy a certain dependence under the renewal process. The conclusion also extend the above Theorem 1 and Theorem 3.1 of Xiao et al. (2013). At the same time, it indicates that tail characteristics of claim distribution determine the ruin probability of insurance company, which is of great significance to the safe operation and the risk assessment of insurance company.

This paper is organized as follows: The second Section introduces the preliminary knowledge. The third Section presents the main results of this paper. The fourth Section gives some lemmas. Finally, the fifth Section gives the proofs of main Theorems.

2. Some Preliminaries

Firstly, we give some markers. All limit relationships of this paper are for $x\to \infty$ unless stated otherwise. For the two positive function $f(·)$ and $g(·)$, if $\lim \sup \frac{f(x)}{g(x)}<\infty$, write $f(x)=O(g(x))$; if $\lim \frac{f(x)}{g(x)}=0$, write $f(x)=o(g(x))$; if $\lim \sup \frac{f(x)}{g(x)}\le 1$, write $f(x)\lesssim g(x)$; if $\mathrm{lim\; inf}\frac{f(x)}{g(x)}\ge 1$, write $f(x)\gtrsim g(x)$; if $\lim \frac{f(x)}{g(x)}$ $=1$, write $f(x)\sim g(x)$; if $f(x)=O(g(x)),g(x)=O(f(x))$, write $f(x)\asymp g(x)$.

Here are some important concepts of heavy-tailed distributions.

Definition 1.

Say a distribution F belongs to the class $\mathcal{L}$, if F satisfies for any $y>0$ (or equivalent for $y=1$)

$$\begin{array}{c}\hfill \underset{x\to \infty }{\lim }\frac{\overline{F}(x+y)}{\overline{F}(x)}=1.\end{array}$$

Say a distribution F belongs to the class $\mathcal{D}$, if F satisfies for any $0<y<1$ (or equivalent for $y=\frac{1}{2}$)

$$\begin{array}{c}\hfill \underset{x\to \infty }{\lim \sup }\frac{\overline{F}(xy)}{\overline{F}(x)}<\infty .\end{array}$$

Say a distribution F belongs to the class $\mathcal{C}$, if F satisfies

$$\begin{array}{c}\hfill \underset{y\uparrow 1}{\lim }\underset{x\to \infty }{\lim \sup }\frac{\overline{F}(xy)}{\overline{F}(x)}=1.\end{array}$$

where their relationship is as follows:

$$\begin{array}{c}\hfill \mathcal{C}\subset \mathcal{L}\cap \mathcal{D}\subset \mathcal{L}.\end{array}$$

For more properties and applications of the heavy-tailed distribution, we can refer to Bingham et al. (1987) and Embrechts et al. (1997).

There are an important relationship between heavy-tailed distribution and $\mathrm{Matuszewska}$ index of the distribution, which is defined by

$$\begin{array}{c}\hfill J_F^{+}=-\underset{x\to \infty }{\lim }\frac{\log {\overline{F}}_{\ast }(x)}{\log x},J_F^{-}=-\underset{x\to \infty }{\lim }\frac{\log {\overline{F}}^{\ast }(x)}{\log x},\end{array}$$

where

$$\begin{array}{c}\hfill {\overline{F}}_{\ast }(y)=\underset{x\to \infty }{\mathrm{lim\; inf}}\frac{\overline{F}(xy)}{\overline{F}(x)},{\overline{F}}^{\ast }(y)=\underset{x\to \infty }{\lim \sup }\frac{\overline{F}(xy)}{\overline{F}(x)}.\end{array}$$

Furthermore, other indices of the distribution F can be defined by

$$\begin{array}{c}\hfill L_F=\underset{y\downarrow 1}{\lim }{\overline{F}}_{\ast }(y).\end{array}$$

As for any $y>0$, there is ${\overline{F}}^{\ast }(y)=1/{\overline{F}}_{\ast }(y)$, hence

$$\begin{array}{c}\hfill L_F=\underset{y\downarrow 1}{\lim }{\overline{F}}_{\ast }(y)=1/\underset{y\uparrow 1}{\lim }{\overline{F}}^{\ast }(y).\end{array}$$

Particularly, if $F\in \mathcal{C}$, then $L_F=1$.

For more properties and applications of the heavy tailed distribution, we can refer to Tang and Tsitsiashvili (2003) and Yang and Wang (2010).

Here we introduce some concepts and properties of dependence.

Definition 2.

If there exists the finite real sequence $\{{\mathrm{g}}_U(n),n\ge 1\}$ for $x_i\in (-\infty ,\infty ),1\le i\le n$ such that

$$\begin{array}{c}\hfill P(\bigcap _{i=1}^n\{{\xi }_i>x_i\})\le {\mathrm{g}}_U(n)\prod _{i=1}^{n}P({\xi }_i>x_i),\end{array}$$

then we say random variable sequence $\{{\xi }_n,n\ge 1\}$ are widely upper orthant dependent $(WUOD)$.

If there exists the finite real sequence $\{{\mathrm{g}}_L(n),n\ge 1\}$ for $x_i\in (-\infty ,\infty ),1\le i\le n$ such that

$$\begin{array}{c}\hfill P(\bigcap _{i=1}^n\{{\xi }_i\le x_i\})\le {\mathrm{g}}_L(n)\prod _{i=1}^{n}P({\xi }_i\le x_i),\end{array}$$

then we say random variable sequence $\{{\xi }_n,n\ge 1\}$ are widely lower orthant dependent $(WLOD)$.

Furthermore, if $\{{\xi }_n,n\ge 1\}$ satisfy $WUOD$ and $WLOD$ at the same time, then we say random variables $\{{\xi }_n,n\ge 1\}$ are widely orthant dependent $(WOD)$.

For more detailed information, we can refer to Wang et al. (2013); Ghosh (1981) and Block et al. (1982).

Definition 3.

If real valued random variables $X_i,i\ge 1$ with distribution functions $F_i,i\ge 1$ satisfy for any $i\ne j$

$$\begin{array}{c}\hfill \underset{x\to \infty }{\lim }P(|X_i|\wedge X_j>x|X_i\vee X_j>x)=0,\end{array}$$

or, equivalently

$$\begin{array}{c}\hfill \underset{x\to \infty }{\lim }\frac{P(X_i>x,X_j>x)+P(X_i<-x,X_j>x)}{{\overline{F}}_i(x)+{\overline{F}}_j(x)}=0,\end{array}$$

then we say random variable variables $X_i,i\ge 1$ are pairwise quasi-asymptotically independent $(PQAI)$.

If real valued random variables $X_i,i\ge 1$ with distribution functions $F_i,i\ge 1$ satisfy for any $i\ne j$

$$\begin{array}{c}\hfill \underset{x_i\wedge x_j\to \infty }{\lim }P(|X_i|>x|X_j>x)=0,\end{array}$$

or, equivalently

$$\begin{array}{c}\hfill \underset{x_i\wedge x_j\to \infty }{\lim }\frac{P(X_i>x,X_j>x)+P(X_i<-x,X_j>x)}{{\overline{F}}_j(x)}=0,\end{array}$$

then we say random variable variables $X_i,i\ge 1$ are pairwise strong quasi-asymptotically independent $(PSQAI$).

Remark 1.

If random variables $X_i,i\ge 1$ are $PSQAI$, then they are $PQAI$.

For more detailed information, we can refer to Li (2013) and Liu et al. (2012).

The first lemma comes from Theorem $2.1$ of Li (2013).

Lemma 1.

Assume that $\{X_j,1\le j\le n\}$ are n real-valued random variables with functions of distribution $F_j,1\le j\le n$. Then

$$\begin{array}{c}\hfill P(\sum _{j=1}^n{c}_j{X}_j>x)\sim \sum _{j=1}^{n}P(c_j{X}_j>x).\end{array}$$

holds if either (i) $\{X_j\}$ are $PSQAI$ and $F_j\in \mathcal{L}\cap \mathcal{D}$ for $1\le j\le n$ and $(c_1,\cdots ,c_n)\in {[a,b]}^n$, or (ii) $\{X_j\}$ are $PQAI$ and $F_j\in \mathcal{C}$ for $1\le j\le n$ and $(c_1,\cdots ,c_n)\in {[a,b]}^n$.

The following lemma comes from Proposition $\mathrm{2.2.1}$ of Bingham et al. (1987).

Lemma 2.

Let $F\in \mathcal{D}$. For any $0<{\alpha }^{\prime }<\alpha <\beta <{\beta }^{\prime }$, there exist positive constants $A_i$ and $B_i,i=1,2$ satisfying the following inequality

$$\begin{array}{c}\hfill \frac{\overline{F}(y)}{\overline{F}(x)}\ge A_1{(x/y)}^{{\alpha }^{\prime }},\end{array}$$

(4)

for any $x\ge y\ge B_1$, and the inequality

$$\begin{array}{c}\hfill \frac{\overline{F}(y)}{\overline{F}(x)}\le A_2{(x/y)}^{{\beta }^{\prime }}.\end{array}$$

(5)

for any $x\ge y\ge B_2$.

The following lemma comes from Theorem $3.3$ of Cline and Samorodnitsky (1994), Lemma $3.4$ of Liu and Wang (2016) and Lemma $3.5$ of Tang and Tsitsiashvili (2003).

Lemma 3.

Let X be a random variable with distribution F and Y be a random variable independent of X. Suppose that H is the distribution of $XY$. If $E{Y}^p<\infty$ for any $0<J_F^{+}\le \beta <{\beta }^{\prime }$ and some $p>J_F^{+}$, then there exist the following conclusions:

(i) 

If $\begin{array}{c}F\in \mathcal{D},then\overline{F}(x)\asymp \overline{H}(x)and{x}^{-{\beta }^{\prime }}=o(\overline{F}(x));\hfill \end{array}$

(ii) 

If $\begin{array}{c}F\in \mathcal{C},H\in \mathcal{C}.\hfill \end{array}$

The following lemma can be proved in Appendix A.

Lemma 4.

$(1)$Under the conditions of Theorem 1 $(\mathrm{or}\mathrm{Theorem}2)$, for all $T\ge T_0$, then we have

$$\begin{array}{c}P({\sum }_{j=1}^{\infty }{Y}_j^{(i)}{e}^{-\delta (S_j^{(i)}+D_j^{(i)})}{I}_{\{S_j^{(i)}+D_j^{(i)}\le T,D_j^{(i)}\le C_j^{(i)}\}}{I}_{\{S_j^{(i)}\le T\}}>x_i)\\ \sim {\sum }_{j=1}^{\infty }P(Y_j^{(i)}{e}^{-\delta (S_j^{(i)}+D_j^{(i)})}{I}_{\{S_j^{(i)}+D_j^{(i)}\le T,D_j^{(i)}\le C_j^{(i)}\}}{I}_{\{S_j^{(i)}\le T\}}>x_i).\end{array}$$

(6)

$(2)$Suppose the conditions of Theorem 2 are true and the inter-arrival times $\{{\theta }_j^i,j\ge 1\}$ are $WLOD$ random variables satisfying $(16)$ for some ${ϵ}_{i0}>0$, depending on $F_i$ and $G_i,i=1,2$. Then, for all $T\ge T_0$, the relation $(6)$ still holds uniformly.

3. Main Results

In this paper, we make the following assumptions:

$A_1$ Assume that the i-th class of random variables $\{C_j^{(i)},j\ge 1\},\{D_j^{(i)},j\ge 1\},\{Y_j^{(i)},j\ge 1\}$, $\{S_j^{(i)},j\ge 1\}$, $i=1,2,$ are independent mutually.

$A_2$ Assume that $E{(N_i(t))}^{p+1}<\infty$ for any fixed $t>0$ and some $p>J_{F_i}^{+},i=1,2$.

Theorem 1.

Consider the bidimensional risk model $(1)$ under the assumptions $A_1-A_2$. Let $p_0=P(S_1^{(i)}<T_0)>0$ for some $0<T_0<\infty$. Assume that claim sizes, $\{Y_j^{(i)},j\ge 1\}$ be $PSQAI$ random variables with common distribution $F_i\in \mathcal{L}\cap \mathcal{D},i=1,2$ such that $J_{F_i}^{-}>0$, respectively. Then for $T\ge T_0$, we have

$$\begin{array}{c}\hfill {\psi }_{\max }(x_1,x_2,T)\sim \prod _{i=1}^2(\sum _{\ell =1}^K{p}_{\ell }{\int }_0^T{\int }_0^{{\alpha }_{\ell }\wedge (T-s)}{\overline{F}}_i(e^{\delta (s+y)}{x}_i)d{H}_i(y)d{m}_i(s)).\end{array}$$

(7)

and

$$\begin{array}{c}\hfill {\psi }_{\min }(x_1,x_2,T)\sim \sum _{i=1}^2\sum _{\ell =1}^K{p}_{\ell }{\int }_0^T{\int }_0^{{\alpha }_{\ell }\wedge (T-s)}{\overline{F}}_i(e^{\delta (s+y)}{x}_i)d{H}_i(y)d{m}_i(s).\end{array}$$

(8)

where $a\wedge b=\min \{a,b\}$.

Proof. 

(i) Firstly, we deal with the relation $(7)$. Write $U_i(t)={\sum }_{j=1}^{N_i(t)}{Y}_j^{(i)}{e}^{-\delta (S_j^{(i)}+D_j^{(i)})}$ $I_{\{S_j^{(i)}+D_j^{(i)}\le t,D_j^{(i)}\le C_j^{(i)}\}}$. Due to $(1)$ and $(2)$, we know that for all $T\ge T_0$,

$$\begin{array}{c}\hfill {\psi }_{\max }(x_1,x_2,T)=P(\bigcap _{i=1}^2\{U_i(s)-\sum _{j=1}^{N_i(t)}{f}_i(C_j^{(i)})e^{-\delta S_j^{(i)}}>x_i\}\mathrm{for}\mathrm{some}0\le s\le T).\end{array}$$

(9)

By $(9)$, we know for all $T\ge T_0$,

$$\begin{array}{c}\hfill P(\bigcap _{i=1}^2\{U_i(T)-\sum _{j=1}^{N_i(T)}{f}_i(C_j^{(i)})e^{-\delta S_j^{(i)}}>x_i\})\le {\psi }_{\max }(x_1,x_2,T)\le P(\bigcap _{i=1}^2\{U_i(T)>x_i\}).\end{array}$$

(10)

Because Lemma $4(1)$ and random variables $\{C_j^{(i)},j\ge 1\},\{D_j^{(i)},j\ge 1\},\{Y_j^{(i)},j\ge 1\}$, $\{S_j^{(i)},j\ge 1\}$, $i=1,2,$ are independent mutually, it is clear that

$$\begin{array}{cc}{\psi }_{\max }(x_1,x_2,T)& \le {\prod }_{i=1}^{2}P(U_i(T)>x_i)\hfill \\ & \sim {\prod }_{i=1}^{2}P({\sum }_{j=1}^{\infty }{Y}_j^{(i)}{e}^{-\delta (S_j^{(i)}+D_j^{(i)})}{I}_{\{S_j^{(i)}+D_j^{(i)}\le T,D_j^{(i)}\le C_j^{(i)}\}}{I}_{\{S_j^{(i)}\le T\}}>x_i)\hfill \\ & \sim {\prod }_{i=1}^2({\sum }_{\ell =1}^K{p}_{\ell }{\int }_0^T{\int }_0^{{\alpha }_{\ell }\wedge (T-s)}{\overline{F}}_i(e^{\delta (s+y)}{x}_i)d{H}_i(y)d{m}_i(s)).\hfill \end{array}$$

(11)

holds uniformly for all $T\ge T_0$.

Because $f_i(C_j^{(i)}),j=1,\cdots ,K$ are bounded and $\{Y_j^{(i)},j\ge 1\}$ are $PSQAI$ random variables with distributions $F_i\in \mathcal{L}\cap \mathcal{D},i=1,2$, respectively, it is easy to prove that $\{Y_j^{(i)}+f_i(C_j^{(i)}),j\ge 1\}$ are $PSQAI$ random variables, whose distributions belong to the class $\mathcal{L}\cap \mathcal{D},i=1,2$, respectively. Hence, by Lemma 4 $(1)$, we know

$$\begin{array}{cc}{\psi }_{\max }(x_1,x_2,T)& \ge P({\bigcap }_{i=1}^2\{U_i(T)-{\sum }_{j=1}^{N_i(T)}{f}_i(C_j^{(i)})e^{-\delta S_j^{(i)}}>x_i\})\hfill \\ & ={\prod }_{i=1}^{2}P(\{U_i(T)-{\sum }_{j=1}^{N_i(T)}{f}_i(C_j^{(i)})e^{-\delta S_j^{(i)}}>x_i\})\hfill \\ & ={\prod }_{i=1}^{2}P({\sum }_{j=1}^{N_i(t)}(Y_j^{(i)}{e}^{-\delta (S_j^{(i)}+D_j^{(i)})}{I}_{\{S_j^{(i)}+D_j^{(i)}\le t,D_j^{(i)}\le C_j^{(i)}\}}-f_i(C_j^{(i)})e^{-\delta S_j^{(i)}})>x_i)\hfill \\ & \sim {\prod }_{i=1}^{2}P({\sum }_{j=1}^{\infty }{Y}_j^{(i)}{e}^{-\delta (S_j^{(i)}+D_j^{(i)})}{I}_{\{S_j^{(i)}+D_j^{(i)}\le t,D_j^{(i)}\le C_j^{(i)}\}}{I}_{\{S_j^{(i)}\le t\}}>x_i)\hfill \\ & \sim {\prod }_{i=1}^2({\sum }_{\ell =1}^K{p}_{\ell }{\int }_0^T{\int }_0^{{\alpha }_{\ell }\wedge (T-s)}{\overline{F}}_i(e^{\delta (s+y)}{x}_i)d{H}_i(y)d{m}_i(s)).\hfill \end{array}$$

(12)

Combining $(11)$ with $(12)$, we obtain that $(7)$ holds uniformly for all $T\ge T_0$.

Next, we handle $(8)$. By $(3)$, we know that

$$\begin{array}{c}\hfill {\psi }_{\min }(x_1,x_2,T)=P({\tau }_1(x_1\le T))+P({\tau }_2(x_2\le T))-{\psi }_{\max }(x_1,x_2,T).\end{array}$$

(13)

where $({\tau }_i(x_i)=\inf \{t:R_i(t)<0|R_i(0)=x_i\},i=1,2$. Then by $(7)$ with its one-dimensional case, it is clear that

$$\begin{array}{c}\hfill P({\tau }_i(x_i)\le T)\sim \sum _{\ell =1}^K{p}_{\ell }{\int }_0^T{\int }_0^{{\alpha }_{\ell }\wedge (T-s)}{\overline{F}}_i(e^{\delta (s+y)}{x}_i)d{H}_i(y)d{m}_i(s),i=1,2.\end{array}$$

(14)

holds uniformly for all $T\ge T_0$.

Again by $(7)$ and $F_i\in \mathcal{L}\cap \mathcal{D}\subset \mathcal{D},i=1,2$, we have that

$$\begin{array}{c}{\lim \sup }_{x_1\wedge x_2\to \infty }{\sup }_{T\in \Lambda \cap [T_0,\infty )}{\displaystyle \frac{{\psi }_{\max }(x_1,x_2,T)}{{\sum }_{i=1}^2{\sum }_{\ell =1}^K{p}_{\ell }{\int }_0^T{\int }_0^{{\alpha }_{\ell }\wedge (T-s)}{\overline{F}}_i(e^{\delta (s+y)}{x}_i)d{H}_i(y)d{m}_{i}s}}\hfill \\ \le {\lim \sup }_{x_1\wedge x_2\to \infty }{\sup }_{T\in \Lambda \cap [T_0,\infty )}{\displaystyle \frac{{\overline{F}}_1(x_1)m_1(T){\overline{F}}_2(x_2)m_2(T){({\sum }_{\ell =1}^K{p}_{\ell }{H}_i({\alpha }_{\ell }\wedge (T-s)))}^2}{{\overline{F}}_1(x_1{e}^{\delta T})m_1(T_0)({\sum }_{\ell =1}^K{p}_{\ell }{H}_i({\alpha }_{\ell }\wedge (T_0-s)))}}\hfill \\ =0.\hfill \end{array}$$

(15)

Hence, by $(13)$–$(15)$, we prove that $(8)$ holds uniformly for all $T\ge T_0$. □

Theorem 2.

Consider the bidimensional risk model $(1)$ under the assumption $A_1-A_2$. Let $p_0=P(S_1^{(i)}<T_0)>0$ for some $0<T_0<\infty$. Assume that claim sizes, $\{Y_i,i\ge 1\}$ be $PQAI$ random variables with common distribution $F_i\in \mathcal{C},i=1,2$ such that $J_{F_i}^{-}>0$, respectively. Then the relations $(7)$ and $(8)$ hold uniformly for $T\ge T_0$.

Proof. 

Similarly, when $f_i(C_j^{(i)}),j=1,\cdots ,K$ are bounded and $\{Y_j^{(i)},j\ge 1\}$ are $PQAI$ random variables with distributions $F_i\in \mathcal{C},i=1,2$, respectively, it is easy to prove that $\{Y_j^{(i)}+f_i(C_j^{(i)}),j\ge 1\}$ are $PQAI$ random variables, whose distributions belong to the class $\mathcal{C},i=1,2$, respectively. Hence, applying the same method of proof of Theorem 1, we know that $(7)$ and $(8)$ still hold uniformly for all $T\ge T_0$. □

Theorem 3.

Under the conditions of Theorem 2, suppose that entry inter-arrival times, $\{{\theta }_k^{(i)},k\ge 1\}$ are $WLOD$ random variables with common distribution $G_i$ satisfying

$$\begin{array}{c}\hfill \underset{n\to \infty }{\lim }{\mathrm{g}}_{L_i}(n)e^{-{ϵ}_{i0}n}=0.\end{array}$$

(16)

holds for some ${ϵ}_{i0}>0$, depending on $F_i$ and $G_i,i=1,2$. Then, the relations $(7)$ and $(8)$ still hold uniformly for all $T\ge T_0$.

Proof. 

By the Lemma $4(2)$, Theorem 2 and similar proof of Theorem 1, the relations $(7)$ and $(8)$ still hold uniformly for all $T\ge T_0$. □

Corollary 1.

Consider one-dimensional risk model satisfying the same conditions as those in Theorem 1, and denote ruin time by $\tau (x)=\inf \{t:R(t)<0|R(0)=x\}$, where

$$\begin{array}{c}\hfill R(t)=x{e}^{\delta t}+\sum _{j=1}^{N(t)}f(C_j)e^{\delta (t-S_j)}-\sum _{j=1}^{N(t)}{Y}_j{e}^{\delta (t-S_j-D_j)}{I}_{\{S_j+D_j\le t,D_j\le C_j\}},\end{array}$$

then, we have

$$\begin{array}{c}\hfill \psi (x,T)=P(\tau (x)\le T)\sim \sum _{\ell =1}^K{p}_{\ell }{\int }_0^T{\int }_0^{{\alpha }_{\ell }\wedge (T-s)}\overline{F}(e^{\delta (s+y)}x)dH(y)dm(s).\end{array}$$

It is easy to prove Corollary 1 from the proof of relation $(7)$ of Theorem 1.

Remark 2.

Corollary 1 is a partial extension for the results of Theorem 3.1 of Xiao et al. (2013), Theorem 1 of Xiao and Tang (2009), and Theorem 3.1 of Xiao and Xie (2018).

4. Conclusions

In summary, this paper studies the two-dimensional independent risk model based on entrance processes with constant interest rate. Under the assumptions that the entry process of policies of two kinds of business of insurance companies have different renewal processes, the claims sizes of two kinds of business are independent of each other, and the claims sizes of the same kind of business are pairwise strong quasi-asymptotically independent, which belong to the class $\mathcal{L}\cap \mathcal{D}$, the maximum finite-time ruin probability and the minimum finite-time ruin probability are obtained, respectively. If intervals of entry time of the policy satisfy the wide lower quadrant dependence, The finite-time maximum ruin probability and the finite-time minimum ruin probability are also obtained.