Claim Sizes-Based Perturbed Risk Model with the Dependence Structure

Abstract

In this paper, we focus on the perturbed risk model with dependent relation and consider the relevance from two aspects. For one side, we use copula function to model the structure of the claim size and interclaim time, and on the other side, we establish the change of premium rat depending on the random thresholds. At last, we obtain the Integro-differential equations and its Laplace transforms of the Gerber-Shiu functions for the new risk model.

Keywords

Share and Cite:

Shen, Y. (2018) Claim Sizes-Based Perturbed Risk Model with the Dependence Structure. Applied Mathematics, 9, 1281-1298. doi: 10.4236/am.2018.911084.

1. Introduction

Before a hundred years ago, the Lundberg-Cram’er classical model laid the foundation for ruin theory. For its fundamental position, we also call it the compound Poisson risk model, and the surplus process of an insurer is denote by

$U\left(t\right)=u+ct-S\left(t\right)$ (1.1)

where $u\ge 0$ denote the initial capital, $c\ge 0$ is the constant premium rate.

$S\left(t\right)=\underset{i=1}{\overset{N\left(t\right)}{\sum }}{X}_{i}$ is the aggregate claims until t, and ${X}_{i}$ is the claim in the i-th

time. $\left\{{X}_{i},i\ge 0\right\}$ is a sequence of independent and identically distributed nonnegative variables with a common probability density function f and probability distribution function is F. $\left\{N\left(t\right),t\ge 0\right\}$ is a Poisson process representing the number of claims in the interval $\left[0,t\right]$ . We define the ultimate ruin probability by

$\psi \left(u\right)=\mathrm{Pr}\left(\tau <\infty |U\left(0\right)=u\right)$ (1.2)

Because of the rapidly development of the financial and insurance industry, scholars have found that adding perturbations to the original model (1.1) can be better reflect the growth pattern of insurance. They explored many ways, such as linear functions, piecewise functions, levy process, jump-diffusion process, etc. Gerber and Shiu put forward a classical function called the Gerber-Shiu expected discounted penalty functions to study ruin probability better, and they use the Brownian motion to be the perturbation term for the first time. The classical model changes to be

$U\left(t\right)=u+ct-S\left(t\right)+\sigma B\left(t\right)$ (1.3)

$B\left(t\right)$ is a standard Brownian motion and it is independent with the aggregate claims process $S\left(t\right)$ , and $\sigma >0$ is the diffusion volatility.

The Gerber-Shiu expected discounted penalty function (EDP) is

$\varphi \left(u\right)=E\left[{\text{e}}^{-\delta \tau }\omega \left(U\left(\tau -\right),|U\left(\tau \right)|\right)I\left(\tau <\infty \right)|U\left(0\right)=u\right],u\ge 0$ (1.4)

where $I\left(\cdot \right)$ is the indicator function, $\omega \left({x}_{1},{x}_{2}\right)$ is a no-negative function of the surplus defined on $\left[0,\infty \right)×\left[0,\infty \right)$ before ruin $U\left(\tau -\right)$ and the deficit at ruin $|U\left(\tau \right)|$ . Let $\delta \ge 0$ be the force of interest.

Since the compound Poisson risk model perturbed (1.3) and EDP function were proposed, it has received a lot of attention, and the EDP function has been studied fully (including the equation of integro-differential, the Laplace transform, analytic solutions, etc.), see e.g.      .

At the beginning, researchers consider structure of the claim sizes and the interclaim time in independent for convenient. With the development of research, the independence assumption above does not accord with the actual situation. So many researchers turn to discuss the risk model with various dependent structures, and they built many dependent structures, see e.g.  -  . Zhang and Yang  use the copula function to construct the dependence and obtain better result for the EDP function.

Meanwhile, the research on premium rate is also put forward with the development of insurance industry. Zhou and Cai  analyze the dependence structure between the premium rate and the claim size for model (1.3). It doesn’t consider the dependence of interclaim time and claim size.

This article is based on the above papers. We study the perturbed risk model as (1.3) and model the dependence structure for two sides. For one side, the interclaim times is dependent with claim sizes by a certain bivariate probability density function, and on the other side, the premiums are depending on claim sizes by the random thresholds $\left\{{Q}_{i},i=1,2,\cdots \right\}$ . In Section 2, we describe the structure. In Section 3, we analyze the affecting to ruin due to claims or perturbation under the model (1.3) and introduce some ruin measures. We show the integro-differential equations in two situations of Gerber-Shiu function satisfied in Section 4. In Section 5, we get the Laplace transforms for the Section 4.

2. Analysis of Dependence Structure

We can use various joint functions to establish dependent structures. In this paper, we use the Farlie-Gumbel-Morgenstern (FGM) copula function to describe the dependence structure. We analyze the former perturbed Poisson risk model and establish new dependence structure based on it. Although the FGM function is relatively simple, it can be applied in a variety of environments controlling the size of $\theta$ . The FGM copula is shown by

${C}_{FGM}\left({u}_{1},{u}_{2}\right)={u}_{1}{u}_{2}+\theta {u}_{1}{u}_{2}\left(1-{u}_{1}\right)\left(1-{u}_{2}\right)$ , $0\le {u}_{1},{u}_{2}\le 1$ (2.1)

where $-1\le \theta \le 1$ . We can see that FGM copula also includes the independence case ( $\theta =0$ ) and allows both negative and positive situations.

We assume $\left\{{V}_{i},i\ge 0\right\}$ is the interclaim times and a sequence of exponential random variables V, the density function $g\left(t\right)=\lambda {\text{e}}^{-\lambda t}$ $\left(\lambda >0\right)$ , and the cumulative distribution function $VG\left(t\right)=1-{\text{e}}^{-\lambda t}$ . The joint distribution function of $\left(X,V\right)$ is

${F}_{X,V}\left(x,t\right)=F\left(x\right)G\left(t\right)+\theta F\left(x\right)G\left(t\right)\left(1-F\left(x\right)\right)\left(1-G\left(t\right)\right)$ (2.2)

And we can get the joint density function of $\left(X,V\right)$ is

${f}_{X,V}\left(x,t\right)=\lambda {\text{e}}^{-\lambda t}f\left(x\right)+\theta \left(2\lambda {\text{e}}^{-2\lambda t}-\lambda {\text{e}}^{-\lambda t}\right)h\left(x\right)$ (2.3)

where $g\left(t\right)=\lambda {\text{e}}^{-\lambda t}$ , $h\left(x\right)=\left(1-2F\left(x\right)\right)f\left(x\right)$ . We can also get the conditional probability density function, that is

${f}_{X|V=t}\left(x\right)=f\left(x\right)+\theta \left(2{\text{e}}^{-\lambda t}-1\right)h\left(x\right)$ (2.4)

Then, we analyze the dependence of the premiums and with claim sizes. Let $\left\{{Q}_{i},i=1,2,\cdots \right\}$ is a set of independent identical distribution random thresholds and independent with $\left\{{X}_{i},i\ge 0\right\}$ , the c.d.f. is $L\left(x\right)$ . If ${X}_{j}$ is larger than ${Q}_{j}$ , we put the insured on the first group, and we assume the time follows an exponential distribution until the next claim(the p.d.f. is ${g}_{1}\left(t\right)={\lambda }_{1}{\text{e}}^{-{\lambda }_{1}t},{\lambda }_{1}>0$ ), and the premium at rate ${c}_{1}$ (> 0); if ${X}_{j}$ is smaller than ${Q}_{j}$ , we put the insured on the second group, the time change exponential parameter (the p.d.f. is ${g}_{2}\left(t\right)={\lambda }_{2}{\text{e}}^{-{\lambda }_{2}t},{\lambda }_{2}>0,{\lambda }_{1}\ne {\lambda }_{2}$ ), and the premium at rate ${c}_{2}$ (> 0). Based on the above assumptions, we denote $C\left(t\right)$ the premium growth function and it is a piecewise function.

So we can establish the surplus process of an insurance company in the way blew

$\begin{array}{c}U\left(t\right)=u+C\left(t\right)-\underset{i=1}{\overset{N\left(t\right)}{\sum }}{X}_{i}+\sigma B\left(t\right)\\ =u+{c}_{1}{\int }_{0}^{t}{I}_{\left\{J\left(s=1\right)\right\}}\text{d}s+{c}_{2}{\int }_{0}^{t}{I}_{\left\{J\left(s=2\right)\right\}}\text{d}s-\underset{i=1}{\overset{N\left(t\right)}{\sum }}{X}_{i}+\sigma B\left(t\right)\end{array}$ (2.5)

$J\left(t\right)$ is represent two groups of insured persons. To guarantee $U\left(t\right)$ has a positive drift, we assume that the model following the next condition

$\frac{{c}_{1}}{{\lambda }_{1}}P\left\{X>Q\right\}+\frac{{c}_{2}}{{\lambda }_{2}}P\left\{X0$ (2.6)

Following the regulation, the insurance company would charge its premium when it is higher than the expected loss amount.

At last, we introduce some functions and formulas that used in this paper.

${\phi }_{f}\left(u\right)={\int }_{u}^{\infty }\omega \left(u,y-u\right)f\left(x\right)\text{d}x$

In this paper, we also use Dickson-Hipp operator ${T}_{r}$ , $r\ge 0$ and some properties following

${T}_{r}f\left(x\right)={\int }_{x}^{\infty }{\text{e}}^{-r\left(y-x\right)}f\left(y\right)\text{d}y$ ,

${T}_{r}f\left(0\right)=\stackrel{˜}{f}\left(r\right),r\ge 0$ .

${T}_{{r}_{1}}{T}_{{r}_{2}}f\left(x\right)={T}_{{r}_{2}}{T}_{{r}_{1}}f\left(x\right)=\frac{{T}_{{r}_{1}}f\left(x\right)-{T}_{{r}_{2}}f\left(x\right)}{{r}_{2}-{r}_{1}},{r}_{1},{r}_{2}\ge 0,{r}_{1}\ne {r}_{2},$

${T}_{{r}_{1}}{T}_{{r}_{2}}f\left(0\right)={T}_{{r}_{2}}{T}_{{r}_{1}}f\left(0\right)=\frac{\stackrel{˜}{f}\left({r}_{1}\right)-\stackrel{˜}{f}\left({r}_{2}\right)}{{r}_{2}-{r}_{1}},{r}_{1},{r}_{2}\ge 0,{r}_{1}\ne {r}_{2}.$

3. The Gerber-Shiu Function

Let ${\tau }_{i}=\mathrm{inf}\left\{t\ge 0,U\left(t\right)<0\right\}$ be the ruin time for $U\left(t\right)$ for the first time under the zero level, and ${\tau }_{i}=\infty$ if $U\left(t\right)\ge 0$ for all $t\ge 0$ .

We analyze the EDP function in two situations in initial surplus u

${\varphi }_{i}\left(u\right)=E\left[{\text{e}}^{-\delta {\tau }_{i}}\omega \left(U\left({\tau }_{i}-\right),|U\left(\tau \right)|\right)I\left({\tau }_{i}<\infty \right)|U\left(0\right)=u\right],u\ge 0,i=1,2$ (3.1)

Because of the perturbation term, we should decompose the EDP function according to two side, that it is whether the ruin is caused by a claim (denote ${\varphi }_{i,w}\left(u\right)$ ) or oscillation (denote ${\varphi }_{i,d}\left(u\right)$ ) And there are four cases for the ruin situation:

${\varphi }_{i}\left(u\right)={\varphi }_{i,d}\left(u\right)+{\varphi }_{i.w}\left(u\right),i=1,2.$ (3.2)

$\begin{array}{l}{\varphi }_{i,w}\left(u\right)=E\left[{\text{e}}^{-\delta {\tau }_{i}}\omega \left(U\left({\tau }_{i}-\right),|U\left(\tau \right)|\right)I\left({\tau }_{i}<\infty ,U\left({\tau }_{i}-\right)<0\right)|U\left(0\right)=u\right],\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,2\end{array}$ (3.3)

$\begin{array}{c}{\varphi }_{i,d}\left(u\right)=E\left[{\text{e}}^{-\delta {\tau }_{i}}\omega \left(U\left({\tau }_{i}-\right),|U\left(\tau \right)|\right)I\left({\tau }_{i}<\infty ,U\left({\tau }_{i}-\right)=0\right)|U\left(0\right)=u\right]\\ =\omega \left(0,0\right)E\left[{\text{e}}^{-\delta {\tau }_{i}}I\left({\tau }_{i}<\infty ,U\left({\tau }_{i}-\right)=0\right)|U\left(0\right)=u\right]\end{array}$ (3.4)

In generally, we suppose that $\omega \left(0,0\right)=1$ . We set a special setting of $\delta =0$ and $\omega =1$ brings ${\varphi }_{i,w}\left(u\right)$ and ${\varphi }_{i,d}\left(u\right)$ to the ruin probabilities.

4. Analysis of the Integro-Differential Equations

In order to discuss the properties of ${\varphi }_{i,w}\left(u\right)$ and ${\varphi }_{i,d}\left(u\right)$ functions, we usually have to get the Integro-differential equations at first.

Now we use a Brownian motion ${W}_{i}\left(t\right)=-{c}_{i}t-\sigma B\left(t\right),i=1,2$ for an auxiliary function, and the drift is $-{c}_{i}$ begin with 0. The variance is ${\sigma }^{2}$ . Let $\stackrel{¯}{{W}_{i}}\left(t\right)={\mathrm{sup}}_{0\le s\le t}{W}_{i}\left(s\right)$ is the supremum in interval $\left[0,t\right]$ . Denoting the first hitting time by ${\tau }_{i}=\mathrm{inf}\left\{t\ge 0:{W}_{i}\left(t\right)=u\right\}$ . Based on the Formula (2.0.1) by Borrodin and Salminen  , we could get for $\delta \ge 0$ ,

$E\left[{\text{e}}^{-\delta {\tau }_{i}}\right]={\text{e}}^{-{\eta }_{i}u}$ , where ${\eta }_{i}=\frac{{c}_{i}}{{\sigma }^{2}}+\sqrt{\frac{2\delta }{{\sigma }^{2}}+\frac{{c}_{i}^{2}}{{\sigma }^{4}}}$ , $i=1,2$ . (4.1)

For $\delta \ge 0$ , we define the following measure

$\begin{array}{l}{P}_{i}\left(u,\text{d}y,\text{d}x\right)=E\left[{\text{e}}^{-\delta V}I\left(\stackrel{¯}{{W}_{i}}\left(V\right)0,x>0,y (4.2)

We denote ${e}_{q}$ by an exponential random variable and the rate $q>0$ . We could first calculate the following measure.

${U}_{q,i}\left(u,\text{d}y\right)=\mathrm{Pr}\left[\stackrel{¯}{{W}_{i}}\left({e}_{q}\right)0,y (4.3)

Next, we introduce two Lemmas in applied probability to obtain the above Formula (4.3).

Lemma 1. Assume that ${e}_{q}$ is independent with $\left\{{W}_{i}\left(t\right)\right\}$ . The following random variables $\stackrel{¯}{{W}_{i}}\left({e}_{q}\right),\stackrel{¯}{{W}_{i}}\left({e}_{q}\right)-{W}_{i}\left({e}_{q}\right),i=1,2$ are independent and have exponentially distributed rates

${v}_{1,i}=\frac{{c}_{i}}{{\sigma }^{2}}+\sqrt{\frac{2q}{{\sigma }^{2}}+\frac{{c}_{i}^{2}}{{\sigma }^{4}}},{v}_{2,i}=-\frac{{c}_{i}}{{\sigma }^{2}}+\sqrt{\frac{2q}{{\sigma }^{2}}+\frac{{c}_{i}^{2}}{{\sigma }^{4}}},i=1,2.$

Respectively, we have for $0\le y ,

$\begin{array}{c}{U}_{q,i}\left(u,\text{d}y\right)={\int }_{x\in \left[y,u\right)}\mathrm{Pr}\left[\stackrel{¯}{{W}_{i}}\left({e}_{q}\right)\in \text{d}x,{W}_{i}\left({e}_{q}\right)-\stackrel{¯}{{W}_{i}}\left({e}_{q}\right)+\stackrel{¯}{{W}_{i}}\left({e}_{q}\right)\in \text{d}y\right]\\ ={\int }_{x\in \left[y,u\right)}{v}_{1,i}{\text{e}}^{-{v}_{1,i}x}{v}_{2,i}{\text{e}}^{-{v}_{2,i}\left(x-y\right)}\text{d}x\text{d}y\\ =\frac{{v}_{1,i}{v}_{2,i}}{{v}_{1,i}+{v}_{2,i}}\left[{\text{e}}^{-{v}_{1,i}y}-{\text{e}}^{-\left({v}_{1,i}+{v}_{2,i}\right)u+{v}_{2,i}y}\right]\text{d}y\end{array}$ (4.4)

We have for $y<0$ ,

$\begin{array}{c}{U}_{q,i}\left(u,\text{d}y\right)={\int }_{x\in \left[0,u\right)}\mathrm{Pr}\left[\stackrel{¯}{{W}_{i}}\left({e}_{q}\right)\in \text{d}x,{W}_{i}\left({e}_{q}\right)-\stackrel{¯}{{W}_{i}}\left({e}_{q}\right)+\stackrel{¯}{{W}_{i}}\left({e}_{q}\right)\in \text{d}y\right]\\ ={\int }_{x\in \left[0,u\right)}{v}_{1,i}{\text{e}}^{-{v}_{1,i}x}{v}_{2,i}{\text{e}}^{-{v}_{2,i}\left(x-y\right)}\text{d}x\text{d}y\\ =\frac{{v}_{1,i}{v}_{2,i}}{{v}_{1,i}+{v}_{2,i}}\left[{\text{e}}^{-{v}_{1,i}y}-{\text{e}}^{-\left({v}_{2,i}+{v}_{2,i}\right)u+{v}_{2,i}y}\right]\text{d}y\end{array}$ (4.5)

For each $u>0$ , ${U}_{q,i}\left(u,\text{d}y\right)$ is absolutely continuous with respect to the Lebesgue measure.

Lemma 2. The measure ${P}_{i}\left(u,\text{d}y,\text{d}x\right)$ has a density in the following

For $0\le y ,

$\begin{array}{l}{p}_{i}\left(u,y,x\right)\\ =\frac{{\lambda }_{i}{\eta }_{1,i}{\eta }_{2,i}}{\left({\lambda }_{i}+\delta \right)\left({\eta }_{1,i}+{\eta }_{2,i}\right)}\left({\text{e}}^{-{\eta }_{1,i}y}-{\text{e}}^{-\left({\eta }_{1,i}+{\eta }_{2,i}\right)u+{\eta }_{2,i}y}\right)\left(f\left(x\right)-\theta h\left(x\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+\frac{2{\lambda }_{i}\theta {\omega }_{1,i}{\omega }_{2,i}}{\left(2{\lambda }_{i}+\delta \right)\left({\omega }_{1,i}+{\omega }_{2,i}\right)}\left({\text{e}}^{-{\omega }_{1,i}y}-{\text{e}}^{-\left({\omega }_{1,i}+{\omega }_{2,i}\right)u+{\omega }_{2,i}y}\right)h\left(x\right)\end{array}$ (4.6)

And $y<0$ ,

$\begin{array}{l}{p}_{i}\left(u,y,x\right)\\ =\frac{{\lambda }_{i}{\eta }_{1,i}{\eta }_{2,i}}{\left({\lambda }_{i}+\delta \right)\left({\eta }_{1,i}+{\eta }_{2,i}\right)}\left({\text{e}}^{{\eta }_{2,i}y}-{\text{e}}^{-\left({\eta }_{1,i}+{\eta }_{2,i}\right)u+{\eta }_{2,i}y}\right)\left(f\left(x\right)-\theta h\left(x\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+\frac{2{\lambda }_{i}\theta {\omega }_{1,i}{\omega }_{2,i}}{\left(2{\lambda }_{i}+\delta \right)\left({\omega }_{1,i}+{\omega }_{2,i}\right)}\left({\text{e}}^{{\omega }_{2,i}y}-{\text{e}}^{-\left({\omega }_{1,i}+{\omega }_{2,i}\right)u+{\omega }_{2,i}y}\right)h\left(x\right)\end{array}$ (4.7)

${\eta }_{1,i}=\frac{{c}_{i}}{{\sigma }^{2}}+\sqrt{\frac{2\left({\lambda }_{i}+\delta \right)}{{\sigma }^{2}}+\frac{{c}_{i}^{2}}{{\sigma }^{4}}},\text{\hspace{0.17em}}{\eta }_{2,i}=-\frac{{c}_{i}}{{\sigma }^{2}}+\sqrt{\frac{2\left({\lambda }_{i}+\delta \right)}{{\sigma }^{2}}+\frac{{c}_{i}^{2}}{{\sigma }^{4}}},\text{\hspace{0.17em}}i=1,2.$

${\omega }_{1,i}=\frac{{c}_{i}}{{\sigma }^{2}}+\sqrt{\frac{2\left(2{\lambda }_{i}+\delta \right)}{{\sigma }^{2}}+\frac{{c}_{i}^{2}}{{\sigma }^{4}}},\text{\hspace{0.17em}}{\omega }_{2,i}=-\frac{{c}_{i}}{{\sigma }^{2}}+\sqrt{\frac{2\left(2{\lambda }_{i}+\delta \right)}{{\sigma }^{2}}+\frac{{c}_{i}^{2}}{{\sigma }^{4}}},\text{\hspace{0.17em}}i=1,2.$

Proof. Take into account the value of V

$\begin{array}{l}{P}_{i}\left(u,\text{d}y,\text{d}x\right)={\int }_{0}^{\infty }{\lambda }_{i}{\text{e}}^{-\left({\lambda }_{i}+\delta \right)t}{f}_{X|V=t}\mathrm{Pr}\left[\stackrel{¯}{{W}_{i}}\left(t\right)

With ${U}_{{\lambda }_{i}+\delta ,i}\left(u,\text{d}y\right)$ and ${U}_{2{\lambda }_{i}+\delta ,i}\left(u,\text{d}y\right)$ , we could get the results.

Now, we could obtain the equations of integro-differential for the Gerber-Shiu functions by using the above results.

Theorem 1. ${\varphi }_{1,w}\left(u\right),{\varphi }_{2,w}\left(u\right)$ satisfies the following integro-differential equation

$\begin{array}{c}{\varphi }_{1,w}\left(u\right)=\frac{{\lambda }_{1}{\eta }_{1,1}{\eta }_{2,1}}{\left({\lambda }_{1}+\delta \right)\left({\eta }_{1,1}+{\eta }_{2,1}\right)}\left({m}_{1,1}\left(u\right)+{T}_{{\eta }_{2,1}}{\gamma }_{1}\left(v\right)-{\text{e}}^{-{\eta }_{1,1}u}{\stackrel{˜}{\gamma }}_{1}\left({\eta }_{2,1}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{2{\lambda }_{1}\theta {\omega }_{1,1}{\omega }_{2,1}}{\left(2{\lambda }_{1}+\delta \right)\left({\omega }_{1,1}+{\omega }_{2,1}\right)}\left({m}_{1,2}\left(u\right)+{T}_{{\omega }_{2,1}}{\gamma }_{2}\left(v\right)-{\text{e}}^{-{\omega }_{1,1}u}{\stackrel{˜}{\gamma }}_{2}\left({\omega }_{2,1}\right)\right)\end{array}$ (4.8)

$\begin{array}{c}{\varphi }_{2,w}\left(u\right)=\frac{{\lambda }_{2}{\eta }_{1,2}{\eta }_{2,2}}{\left({\lambda }_{2}+\delta \right)\left({\eta }_{1,2}+{\eta }_{2,2}\right)}\left({m}_{2,1}\left(u\right)+{T}_{{\eta }_{2,2}}{\gamma }_{1}\left(v\right)-{\text{e}}^{-{\eta }_{1,2}u}\stackrel{˜}{\gamma }\left({\eta }_{2,2}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{2{\lambda }_{2}\theta {\omega }_{1,2}{\omega }_{2,2}}{\left(2{\lambda }_{2}+\delta \right)\left({\omega }_{1,2}+{\omega }_{2,2}\right)}\left({m}_{2,2}\left(u\right)+{T}_{{\omega }_{2,2}}{\gamma }_{2}\left(v\right)-{\text{e}}^{-{\omega }_{1,2}u}{\stackrel{˜}{\gamma }}_{2}\left({\omega }_{2,2}\right)\right)\end{array}$ (4.9)

${m}_{i,1}\left(u\right)={\int }_{0}^{u}{\text{e}}^{-{\eta }_{1,i}\left(u-v\right)}{\gamma }_{1}\left(v\right)\text{d}v,i=1,2$ ,

${m}_{i,2}\left(u\right)={\int }_{0}^{u}{\text{e}}^{-{\omega }_{1,i}\left(u-v\right)}{\gamma }_{2}\left(v\right)\text{d}v,i=1,2$ .

Proof. At first, we analyze the time and size for the first claim, and utilize the proofed ${p}_{i}\left(u,y,x\right)$ , we have

$\begin{array}{c}{\varphi }_{1,w}\left(u\right)={\int }_{t\in \left(0,\infty \right)}{\int }_{y\in \left(-\infty ,u\right)}{\lambda }_{1}{\text{e}}^{-\left({\lambda }_{1}+\delta \right)t}\mathrm{Pr}\left[\stackrel{¯}{{W}_{1}}\left(t\right){Q}_{1}\right){\varphi }_{1,w}\left(u-y-x\right)+P\left(x<{Q}_{1}\right){\varphi }_{2,w}\left(u-y-x\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}×{f}_{X,V}\left(x,t\right)\text{d}x\text{d}t+{\int }_{t\in \left(0,\infty \right)}{\int }_{y\in \left(-\infty ,u\right)}{\lambda }_{1}{\text{e}}^{-\left({\lambda }_{1}+\delta \right)t}\mathrm{Pr}\left[\stackrel{¯}{{W}_{1}}\left(t\right)

By lemma 2, we can get the following form

$\begin{array}{l}{\varphi }_{1,w}\left(u\right)={\int }_{-\infty }^{u}{\int }_{0}^{u-y}\left[P\left(x>{Q}_{1}\right){\varphi }_{1,w}\left(u-y-x\right)+P\left(x<{Q}_{1}\right){\varphi }_{2,w}\left(u-y-x\right)\right]\\ \text{ }×{p}_{1}\left(u,x,y\right)\text{d}x\text{d}y+{\int }_{-\infty }^{u}{\int }_{u-y}^{\infty }\omega \left(u-y,x-\left(u-y\right)\right){p}_{1}\left(u,x,y\right)\text{d}x\text{d}y\end{array}$ (4.10)

And

$\begin{array}{l}{\varphi }_{2,w}\left(u\right)={\int }_{t\in \left(0,\infty \right)}{\int }_{y\in \left(-\infty ,u\right)}{\lambda }_{2}{\text{e}}^{-\left({\lambda }_{2}+\delta \right)t}\mathrm{Pr}\left[\stackrel{¯}{{W}_{2}}\left(t\right){Q}_{1}\right){\varphi }_{1,w}\left(u-y-x\right)+P\left(x<{Q}_{1}\right){\varphi }_{2,w}\left(u-y-x\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }×{f}_{X,V}\left(x,t\right)\text{d}x\text{d}t+{\int }_{t\in \left(0,\infty \right)}{\int }_{y\in \left(-\infty ,u\right)}{\lambda }_{2}{e}^{-\left({\lambda }_{2}+\delta \right)t}\mathrm{Pr}\left[\stackrel{¯}{{W}_{2}}\left(t\right){Q}_{1}\right){\varphi }_{1,w}\left(u-y-x\right)+P\left(x<{Q}_{1}\right){\varphi }_{2,w}\left(u-y-x\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }×{p}_{2}\left(u,x,y\right)\text{d}x\text{d}y+{\int }_{-\infty }^{u}{\int }_{u-y}^{\infty }\omega \left(u-y,x-\left(u-y\right)\right){p}_{2}\left(u,x,y\right)\text{d}x\text{d}y\end{array}$ (4.11)

Let $L\left(x\right)=P\left(x>{Q}_{1}\right),\stackrel{¯}{L}\left(x\right)=1-L\left(x\right)$ , then

$\begin{array}{l}{\varphi }_{1,w}\left(u\right)={\int }_{-\infty }^{0}{\int }_{0}^{u-y}\left[L\left(x\right){\varphi }_{1,w}\left(u-y-x\right)+\stackrel{¯}{L}\left(x\right){\varphi }_{2,w}\left(u-y-x\right)\right]{p}_{1}\left(u,x,y\right)\text{d}x\text{d}y\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{\int }_{0}^{u}{\int }_{0}^{u-y}\left[L\left(x\right){\varphi }_{1,w}\left(u-y-x\right)+\stackrel{¯}{L}\left(x\right){\varphi }_{2,w}\left(u-y-x\right)\right]{p}_{1}\left(u,x,y\right)\text{d}x\text{d}y\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{\int }_{-\infty }^{0}{\int }_{u-y}^{\infty }\omega \left(u-y,x-\left(u-y\right)\right){p}_{1}\left(u,x,y\right)\text{d}x\text{d}y\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{\int }_{0}^{u}{\int }_{u-y}^{\infty }\omega \left(u-y,x-\left(u-y\right)\right){p}_{1}\left(u,x,y\right)\text{d}x\text{d}y\end{array}$

$\begin{array}{l}=\frac{{\lambda }_{1}{\eta }_{1,1}{\eta }_{2,1}}{\left({\lambda }_{1}+\delta \right)\left({\eta }_{1,1}+{\eta }_{2,1}\right)}\left\{{\int }_{0}^{u}{\int }_{0}^{u-y}\left[L\left(x\right){\varphi }_{1,w}\left(u-y-x\right)+\stackrel{¯}{L}\left(x\right){\varphi }_{2,w}\left(u-y-x\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}×\left({\text{e}}^{-{\eta }_{1,1}y}-{\text{e}}^{-\left({\eta }_{1,1}+{\eta }_{2,1}\right)u+{\eta }_{2,1}y}\right)\left(f\left(x\right)-\theta h\left(x\right)\right)\text{d}x\text{d}y\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{0}^{u}{\int }_{u-y}^{\infty }\omega \left(u-y,x-\left(u-y\right)\right)\left({\text{e}}^{-{\eta }_{1,1}y}-{\text{e}}^{-\left({\eta }_{1,1}+{\eta }_{2,1}\right)u+{\eta }_{2,1}y}\right)\left(f\left(x\right)-\theta h\left(x\right)\right)\text{d}x\text{d}y\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{-\infty }^{0}{\int }_{0}^{u-y}\left[L\left(x\right){\varphi }_{1,w}\left(u-y-x\right)+\stackrel{¯}{L}\left(x\right){\varphi }_{2,w}\left(u-y-x\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}×\left({\text{e}}^{{\eta }_{2,1}y}-{\text{e}}^{-\left({\eta }_{1,1}+{\eta }_{2,1}\right)u+{\eta }_{2,1}y}\right)\left(f\left(x\right)-\theta h\left(x\right)\right)\text{d}x\text{d}y\end{array}$

$\begin{array}{l}\text{ }+{\int }_{-\infty }^{0}{\int }_{u-y}^{\infty }\omega \left(u-y,x-\left(u-y\right)\right)\left({\text{e}}^{{\eta }_{2,1}y}-{\text{e}}^{-\left({\eta }_{1,1}+{\eta }_{2,1}\right)u+{\eta }_{2,1}y}\right)\left(f\left(x\right)-\theta h\left(x\right)\right)\text{d}x\text{d}y\right\}\\ \text{ }+\frac{2{\lambda }_{1}\theta {\omega }_{1,1}{\omega }_{2,1}}{\left(2{\lambda }_{1}+\delta \right)\left({\omega }_{1,1}+{\omega }_{2,1}\right)}\left\{{\int }_{0}^{u}{\int }_{0}^{u-y}\left[L\left(x\right){\varphi }_{1,w}\left(u-y-x\right)+\stackrel{¯}{L}\left(x\right){\varphi }_{2,w}\left(u-y-x\right)\right]\\ \text{ }×\left({\text{e}}^{-{\omega }_{1,1}y}-{\text{e}}^{-\left({\omega }_{1,1}+{\omega }_{2,1}\right)u+{\omega }_{2,1}y}\right)h\left(x\right)\text{d}x\text{d}y\\ \text{ }+{\int }_{0}^{u}{\int }_{u-y}^{\infty }\omega \left(u-y,x-\left(u-y\right)\right)\left({\text{e}}^{-{\omega }_{1,1}y}-{\text{e}}^{-\left({\omega }_{1,1}+{\omega }_{2,1}\right)u+{\omega }_{2,1}y}\right)h\left(x\right)\text{d}x\text{d}y\end{array}$

$\begin{array}{l}\text{ }+{\int }_{-\infty }^{0}{\int }_{0}^{u-y}\left[L\left(x\right){\varphi }_{1,w}\left(u-y-x\right)+\stackrel{¯}{L}\left(x\right){\varphi }_{2,w}\left(u-y-x\right)\right]\\ \text{ }×\left({\text{e}}^{{\omega }_{2,1}y}-{\text{e}}^{-\left({\omega }_{1,1}+{\omega }_{2,1}\right)u+{\omega }_{2,1}y}\right)h\left(x\right)\text{d}x\text{d}y\\ \text{ }+{\int }_{-\infty }^{0}{\int }_{u-y}^{\infty }\omega \left(u-y,x-\left(u-y\right)\right)\left({\text{e}}^{{\omega }_{2,1}y}-{\text{e}}^{-\left({\omega }_{1,1}+{\omega }_{2,1}\right)u+{\omega }_{2,1}y}\right)h\left(x\right)\text{d}x\text{d}y\right\}\end{array}$ (4.12)

In the same way,

$\begin{array}{l}{\varphi }_{2,w}\left(u\right)\\ =\frac{{\lambda }_{2}{\eta }_{1,2}{\eta }_{2,2}}{\left({\lambda }_{2}+\delta \right)\left({\eta }_{1,2}+{\eta }_{2,2}\right)}\left\{{\int }_{0}^{u}{\int }_{0}^{u-y}\left[L\left(x\right){\varphi }_{1,w}\left(u-y-x\right)+\stackrel{¯}{L}\left(x\right){\varphi }_{2,w}\left(u-y-x\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}×\left({\text{e}}^{-{\eta }_{12}y}-{\text{e}}^{-\left({\eta }_{1,2}+{\eta }_{2,2}\right)u+{\eta }_{2,2}y}\right)\left(f\left(x\right)-\theta h\left(x\right)\right)\text{d}x\text{d}y\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{0}^{u}{\int }_{u-y}^{\infty }\omega \left(u-y,x-\left(u-y\right)\right)\left({\text{e}}^{-{\eta }_{1,2}y}-{\text{e}}^{-\left({\eta }_{1,2}+{\eta }_{2,2}\right)u+{\eta }_{2,2}y}\right)\left(f\left(x\right)-\theta h\left(x\right)\right)\text{d}x\text{d}y\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{-\infty }^{0}{\int }_{0}^{u-y}\left[L\left(x\right){\varphi }_{1,w}\left(u-y-x\right)+\stackrel{¯}{L}\left(x\right){\varphi }_{2,w}\left(u-y-x\right)\right]\end{array}$

$\begin{array}{l}\text{ }×\left({\text{e}}^{-{\eta }_{2,2}y}-{\text{e}}^{-\left({\eta }_{1,2}+{\eta }_{2,2}\right)u+{\eta }_{2,2}y}\right)\left(f\left(x\right)-\theta h\left(x\right)\right)\text{d}x\text{d}y\\ \text{ }+{\int }_{-\infty }^{0}{\int }_{u-y}^{\infty }\omega \left(u-y,x-\left(u-y\right)\right)\left({\text{e}}^{-{\eta }_{2,2}y}-{\text{e}}^{-\left({\eta }_{1,2}+{\eta }_{2,2}\right)u+{\eta }_{2,2}y}\right)\left(f\left(x\right)-\theta h\left(x\right)\right)\text{d}x\text{d}y\right\}\\ \text{ }+\frac{2{\lambda }_{2}\theta {\omega }_{1,2}{\omega }_{2,2}}{\left(2{\lambda }_{2}+\delta \right)\left({\omega }_{1,2}+{\omega }_{2,2}\right)}\left\{{\int }_{0}^{u}{\int }_{0}^{u-y}\left[L\left(x\right){\varphi }_{1,w}\left(u-y-x\right)\\ \text{ }+\stackrel{¯}{L}\left(x\right){\varphi }_{2,w}\left(u-y-x\right)\right]\left({\text{e}}^{-{\omega }_{1,2}y}-{\text{e}}^{-\left({\omega }_{1,2}+{\omega }_{2,2}\right)u+{\omega }_{2,2}y}\right)h\left(x\right)\text{d}x\text{d}y\end{array}$

$\begin{array}{l}\text{ }+{\int }_{0}^{u}{\int }_{u-y}^{\infty }\omega \left(u-y,x-\left(u-y\right)\right)\left({\text{e}}^{-{\omega }_{1,2}y}-{\text{e}}^{-\left({\omega }_{1,2}+{\omega }_{2,2}\right)u+{\omega }_{2,2}y}\right)h\left(x\right)\text{d}x\text{d}y\\ \text{ }+{\int }_{-\infty }^{0}{\int }_{0}^{u-y}\left[L\left(x\right){\varphi }_{1,w}\left(u-y-x\right)+\stackrel{¯}{L}\left(x\right){\varphi }_{2,w}\left(u-y-x\right)\right]\\ \text{ }×\left({\text{e}}^{{\omega }_{2,2}y}-{\text{e}}^{-\left({\omega }_{1,2}+{\omega }_{2,2}\right)u+{\omega }_{2,2}y}\right)h\left(x\right)\text{d}x\text{d}y\\ \text{ }+{\int }_{-\infty }^{0}{\int }_{u-y}^{\infty }\omega \left(u-y,x-\left(u-y\right)\right)\left({\text{e}}^{{\omega }_{2,2}y}-{\text{e}}^{-\left({\omega }_{1,2}+{\omega }_{2,2}\right)u+{\omega }_{2,2}y}\right)h\left(x\right)\text{d}x\text{d}y\right\}\end{array}$ (4.13)

Let

$\begin{array}{l}{\chi }_{1}\left(x\right)=\stackrel{¯}{L}\left(x\right)\left(f\left(x\right)-\theta h\left(x\right)\right),\\ {\xi }_{1}\left(x\right)=L\left(x\right)\left(f\left(x\right)-\theta h\left(x\right)\right)=\left(f\left(x\right)-\theta h\left(x\right)\right)-{\chi }_{1}\left(x\right)\end{array}$

${\chi }_{2}\left(x\right)=\stackrel{¯}{L}\left(x\right)h\left(x\right),{\xi }_{2}\left(x\right)=L\left(x\right)h\left(x\right)=h\left(x\right)-{\chi }_{2}\left(x\right)$

so

$\begin{array}{l}{\varphi }_{1,w}\left(u\right)=\frac{{\lambda }_{1}{\eta }_{1,1}{\eta }_{2,1}}{\left({\lambda }_{1}+\delta \right)\left({\eta }_{1,1}+{\eta }_{2,1}\right)}\left\{{\int }_{0}^{u}{\int }_{0}^{u-y}\left[{\xi }_{1}\left(x\right){\varphi }_{1,w}\left(u-y-x\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{\chi }_{1}\left(x\right){\varphi }_{2,w}\left(u-y-x\right)\right]×\left({\text{e}}^{-{\eta }_{1,1}y}-{\text{e}}^{-\left({\eta }_{1,1}+{\eta }_{2,1}\right)u+{\eta }_{2,1}y}\right)\text{d}x\text{d}y\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{\int }_{0}^{u}{\int }_{u-y}^{\infty }\omega \left(u-y,x-\left(u-y\right)\right)\left({\text{e}}^{-{\eta }_{1,1}y}-{\text{e}}^{-\left({\eta }_{1,1}+{\eta }_{2,1}\right)u+{\eta }_{2,1}y}\right)\left(f\left(x\right)-\theta h\left(x\right)\right)\text{d}x\text{d}y\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{\int }_{-\infty }^{0}{\int }_{0}^{u-y}\left[{\xi }_{1}\left(x\right){\varphi }_{1,w}\left(u-y-x\right)+{\chi }_{1}\left(x\right){\varphi }_{2,w}\left(u-y-x\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }×\left({\text{e}}^{{\eta }_{2,1}y}-{\text{e}}^{-\left({\eta }_{1,1}+{\eta }_{2,1}\right)u+{\eta }_{2,1}y}\right)\text{d}x\text{d}y+{\int }_{-\infty }^{0}{\int }_{u-y}^{\infty }\omega \left(u-y,x-\left(u-y\right)\right)\end{array}$

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }×\left({\text{e}}^{{\eta }_{2,1}y}-{\text{e}}^{-\left({\eta }_{1,1}+{\eta }_{2,1}\right)u+{\eta }_{2,1}y}\right)\left(f\left(x\right)-\theta h\left(x\right)\right)\text{d}x\text{d}y\right\}\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{2{\lambda }_{1}\theta {\omega }_{1,1}{\omega }_{2,1}}{\left(2{\lambda }_{1}+\delta \right)\left({\omega }_{1,1}+{\omega }_{2,1}\right)}\left\{{\int }_{0}^{u}{\int }_{0}^{u-y}\left[{\xi }_{2}\left(x\right){\varphi }_{1,w}\left(u-y-x\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{\chi }_{2}\left(x\right){\varphi }_{2,w}\left(u-y-x\right)\right]\left({\text{e}}^{-{\omega }_{1,1}y}-{\text{e}}^{-\left({\omega }_{1,1}+{\omega }_{2,1}\right)u+{\omega }_{2,1}y}\right)\text{d}x\text{d}y\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{\int }_{0}^{u}{\int }_{u-y}^{\infty }\omega \left(u-y,x-\left(u-y\right)\right)\left({\text{e}}^{-{\omega }_{1,1}y}-{\text{e}}^{-\left({\omega }_{1,1}+{\omega }_{2,1}\right)u+{\omega }_{2,1}y}\right)h\left(x\right)\text{d}x\text{d}y\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{\int }_{-\infty }^{0}{\int }_{0}^{u-y}\left[{\xi }_{2}\left(x\right){\varphi }_{1,w}\left(u-y-x\right)+{\chi }_{2}\left(x\right){\varphi }_{2,w}\left(u-y-x\right)\right]\end{array}$

$\begin{array}{l}\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}×\left({\text{e}}^{{\omega }_{2,1}y}-{\text{e}}^{-\left({\omega }_{1,1}+{\omega }_{2,1}\right)u+{\omega }_{2,1}y}\right)\text{d}x\text{d}y\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{-\infty }^{0}{\int }_{u-y}^{\infty }\omega \left(u-y,x-\left(u-y\right)\right)\left({\text{e}}^{{\omega }_{2,1}y}-{\text{e}}^{-\left({\omega }_{1,1}+{\omega }_{2,1}\right)u+{\omega }_{2,1}y}\right)h\left(x\right)\text{d}x\text{d}y\right\}\\ =\frac{{\lambda }_{1}{\eta }_{1,1}{\eta }_{2,1}}{\left({\lambda }_{1}+\delta \right)\left({\eta }_{1,1}+{\eta }_{2,1}\right)}\left\{{\int }_{0}^{u}\left({\text{e}}^{-{\eta }_{1,1}y}-{\text{e}}^{-\left({\eta }_{1,1}+{\eta }_{2,1}\right)u+{\eta }_{2,1}y}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }×{\int }_{0}^{u-y}\left(\left({\xi }_{1}\left(x\right){\varphi }_{1,w}\left(u-y-x\right)+{\chi }_{1}\left(x\right){\varphi }_{2,w}\left(u-y-x\right)\right)\text{d}x+{\phi }_{f-\theta h}\left(u-y\right)\right)\text{d}y\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{-\infty }^{0}\left({\text{e}}^{{\eta }_{2,1}y}-{\text{e}}^{-\left({\eta }_{1,1}+{\eta }_{2,1}\right)u+{\eta }_{2,1}y}\right){\int }_{0}^{u-y}\left(\left({\xi }_{1}\left(x\right){\varphi }_{1,w}\left(u-y-x\right)\end{array}$

$\begin{array}{l}\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\chi }_{1}\left(x\right){\varphi }_{2,w}\left(u-y-x\right)\right)\text{d}x+{\phi }_{f-\theta h}\left(u-y\right)\right)\text{d}y\right\}\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{2{\lambda }_{1}\theta {\omega }_{1,1}{\omega }_{2,1}}{\left(2{\lambda }_{1}+\delta \right)\left({\omega }_{1,1}+{\omega }_{2,1}\right)}\left\{{\int }_{0}^{u}\left({\text{e}}^{-{\omega }_{1,1}y}-{\text{e}}^{-\left({\omega }_{1,1}+{\omega }_{2,1}\right)u+{\omega }_{2,1}y}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }×{\int }_{0}^{u-y}\left(\left({\xi }_{2}\left(x\right){\varphi }_{1,w}\left(u-y-x\right)+{\chi }_{2}\left(x\right){\varphi }_{2,w}\left(u-y-x\right)\right)\text{d}x+{\varphi }_{h}\left(u-y\right)\right)\text{d}y\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{\int }_{-\infty }^{0}\left({\text{e}}^{{\omega }_{2,1}y}-{\text{e}}^{-\left({\omega }_{1,1}+{\omega }_{2,1}\right)u+{\omega }_{2,1}y}\right){\int }_{0}^{u-y}\left(\left({\xi }_{2}\left(x\right){\varphi }_{1,w}\left(u-y-x\right)\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\chi }_{2}\left(x\right){\varphi }_{2,w}\left(u-y-x\right)\right)\text{d}x+{\phi }_{h}\left(u-y\right)\right)\text{d}y\right\}\end{array}$ (4.14)

We bring the change of variable $v=u-y$ into the above equation

$\begin{array}{l}{\varphi }_{1,w}\left(u\right)=\frac{{\lambda }_{1}{\eta }_{1,1}{\eta }_{2,1}}{\left({\lambda }_{1}+\delta \right)\left({\eta }_{1,1}+{\eta }_{2,1}\right)}\left\{{\int }_{0}^{u}\left({\text{e}}^{-{\eta }_{1,1}\left(u-v\right)}-{\text{e}}^{-{\eta }_{1,1}u-{\eta }_{2,1}v}\right)\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}×{\int }_{0}^{v}\left(\left({\xi }_{1}\left(x\right){\varphi }_{1,w}\left(v-x\right)+{\chi }_{1}\left(x\right){\varphi }_{2,w}\left(v-x\right)\right)\text{d}x+{\phi }_{f-\theta h}\left(v\right)\right)\text{d}v\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{\int }_{u}^{\infty }\left({\text{e}}^{{\eta }_{2,1}\left(u-v\right)}-{\text{e}}^{-{\eta }_{1,1}u-{\eta }_{2,1}v}\right){\int }_{0}^{v}\left(\left({\xi }_{1}\left(x\right){\varphi }_{1,w}\left(v-x\right)+{\chi }_{1}\left(x\right){\varphi }_{2,w}\left(v-x\right)\right)\text{d}x\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\phi }_{f-\theta h}\left(v\right)\right)\text{d}v\right\}+\frac{2{\lambda }_{1}\theta {\omega }_{1,1}{\omega }_{2,1}}{\left(2{\lambda }_{1}+\delta \right)\left({\omega }_{1,1}+{\omega }_{2,1}\right)}\left\{{\int }_{0}^{u}\left({\text{e}}^{-{\omega }_{1,1}\left(u-v\right)}-{\text{e}}^{-{\omega }_{1,1}u-{\omega }_{2,1}v}\right)\end{array}$

$\begin{array}{l}\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}×{\int }_{0}^{v}\left(\left({\xi }_{2}\left(x\right){\varphi }_{1,w}\left(v-x\right)+{\chi }_{2}\left(x\right){\varphi }_{2,w}\left(v-x\right)\right)\text{d}x+{\phi }_{h}\left(v\right)\right)\text{d}v\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{u}^{\infty }\left({\text{e}}^{{\omega }_{2,1}\left(u-v\right)}-{\text{e}}^{-{\omega }_{1,1}u-{\omega }_{2,1}v}\right){\int }_{0}^{v}\left(\left({\xi }_{2}\left(x\right){\varphi }_{1,w}\left(v-x\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{\chi }_{2}\left(x\right){\varphi }_{2,w}\left(v-x\right)\right)\text{d}x+{\phi }_{h}\left(v\right)\right)\text{d}v\right\}\end{array}$ (4.15)

$\begin{array}{l}{\varphi }_{2,w}\left(u\right)=\frac{{\lambda }_{2}{\eta }_{1,2}{\eta }_{2,2}}{\left({\lambda }_{2}+\delta \right)\left({\eta }_{1,2}+{\eta }_{2,2}\right)}\left\{{\int }_{0}^{u}\left({\text{e}}^{-{\eta }_{12}\left(u-v\right)}-{\text{e}}^{-{\eta }_{1,2}u-{\eta }_{2,2}v}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }×{\int }_{0}^{v}\left(\left({\xi }_{1}\left(x\right){\varphi }_{1,w}\left(v-x\right)+{\chi }_{1}\left(x\right){\varphi }_{2,w}\left(v-x\right)\right)\text{d}x+{\phi }_{f-\theta h}\left(v\right)\right)\text{d}y\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{\int }_{u}^{\infty }\left({\text{e}}^{-{\eta }_{2,2}\left(u-v\right)}-{\text{e}}^{-{\eta }_{1,2}u-{\eta }_{2,2}v}\right){\int }_{0}^{v}\left(\left({\xi }_{1}\left(x\right){\varphi }_{1,w}\left(v-x\right)+{\chi }_{1}\left(x\right){\varphi }_{2,w}\left(v-x\right)\right)\text{d}x\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{\phi }_{f-\theta h}\left(v\right)\right)\text{d}v\right\}+{\int }_{u}^{\infty }\left({\text{e}}^{{\omega }_{2,2}\left(u-v\right)}-{\text{e}}^{-{\omega }_{1,2}u-{\omega }_{2,2}v}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }×{\int }_{0}^{v}\left(\left({\xi }_{2}\left(x\right){\varphi }_{1,w}\left(v-x\right)+{\chi }_{2}\left(x\right){\varphi }_{2,w}\left(v-x\right)\right)\text{d}x+{\phi }_{h}\left(v\right)\right)\text{d}v\right\}\end{array}$ (4.16)

Let

${\gamma }_{1}\left(y\right)={\int }_{0}^{y}\left({\xi }_{1}\left(x\right){\varphi }_{1,w}\left(y-x\right)+{\chi }_{1}\left(x\right){\varphi }_{2,w}\left(y-x\right)\right)\text{d}x+{\phi }_{f-\theta h}\left(y\right)$ ,

${\gamma }_{2}\left(y\right)={\int }_{0}^{y}\left({\xi }_{2}\left(x\right){\varphi }_{1,w}\left(y-x\right)+{\chi }_{2}\left(x\right){\varphi }_{2,w}\left(y-x\right)\right)\text{d}x+{\phi }_{h}\left(y\right)$ .

We can rewrite the equation

$\begin{array}{c}{\varphi }_{1,w}\left(u\right)=\frac{{\lambda }_{1}{\eta }_{1,1}{\eta }_{2,1}}{\left({\lambda }_{1}+\delta \right)\left({\eta }_{1,1}+{\eta }_{2,1}\right)}\left\{{\int }_{0}^{u}\left({\text{e}}^{-{\eta }_{1,1}\left(u-v\right)}-{\text{e}}^{-{\eta }_{1,1}u-{\eta }_{2,1}v}\right){\gamma }_{1}\left(v\right)\text{d}v\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{\int }_{u}^{\infty }\left({\text{e}}^{{\eta }_{2,1}\left(u-v\right)}-{\text{e}}^{-{\eta }_{1,1}u-{\eta }_{2,1}v}\right){\gamma }_{1}\left(v\right)\text{d}v\right\}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+\frac{2{\lambda }_{1}\theta {\omega }_{1,1}{\omega }_{2,1}}{\left(2{\lambda }_{1}+\delta \right)\left({\omega }_{1,1}+{\omega }_{2,1}\right)}\left\{{\int }_{0}^{u}\left({\text{e}}^{-{\omega }_{1,1}\left(u-v\right)}-{\text{e}}^{-{\omega }_{1,1}u-{\omega }_{2,1}v}\right){\gamma }_{2}\left(v\right)\text{d}v\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{\int }_{u}^{\infty }\left({\text{e}}^{{\omega }_{2,1}\left(u-v\right)}-{\text{e}}^{-{\omega }_{1,1}u-{\omega }_{2,1}v}\right){\gamma }_{2}\left(v\right)\text{d}v\right\}\end{array}$

$\begin{array}{l}=\frac{{\lambda }_{1}{\eta }_{1,1}{\eta }_{2,1}}{\left({\lambda }_{1}+\delta \right)\left({\eta }_{1,1}+{\eta }_{2,1}\right)}\left\{{\int }_{0}^{u}{\text{e}}^{-{\eta }_{1,1}\left(u-v\right)}{\gamma }_{1}\left(v\right)\text{d}v\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{u}^{\infty }{\text{e}}^{{\eta }_{2,1}\left(u-v\right)}{\gamma }_{1}\left(v\right)\text{d}v-{\int }_{0}^{\infty }{\text{e}}^{-{\eta }_{1,1}u-{\eta }_{2,1}v}{\gamma }_{1}\left(v\right)\text{d}v\right\}\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{2{\lambda }_{1}\theta {\omega }_{1,1}{\omega }_{2,1}}{\left(2{\lambda }_{1}+\delta \right)\left({\omega }_{1,1}+{\omega }_{2,1}\right)}\left\{{\int }_{0}^{u}{\text{e}}^{-{\omega }_{1,1}\left(u-v\right)}{\gamma }_{2}\left(v\right)\text{d}v\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{u}^{\infty }{\text{e}}^{{\omega }_{2,1}\left(u-v\right)}{\gamma }_{2}\left(v\right)\text{d}v-{\int }_{0}^{\infty }{\text{e}}^{-{\omega }_{1,1}u-{\omega }_{2,1}v}{\gamma }_{2}\left(v\right)\text{d}v\right\}\end{array}$ (4.17)

$\begin{array}{c}{\varphi }_{2,w}\left(u\right)=\frac{{\lambda }_{2}{\eta }_{1,2}{\eta }_{2,2}}{\left({\lambda }_{2}+\delta \right)\left({\eta }_{1,2}+{\eta }_{2,2}\right)}\left\{{\int }_{0}^{u}\left({\text{e}}^{-{\eta }_{1,2}\left(u-v\right)}-{\text{e}}^{-{\eta }_{1,2}u-{\eta }_{2,2}v}\right){\gamma }_{1}\left(v\right)\text{d}v\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{\int }_{u}^{\infty }\left({\text{e}}^{-{\eta }_{2,2}\left(u-v\right)}-{\text{e}}^{-{\eta }_{1,2}u-{\eta }_{2,2}v}\right){\gamma }_{1}\left(v\right)\text{d}v\right\}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+\frac{2{\lambda }_{2}\theta {\omega }_{1,2}{\omega }_{2,2}}{\left(2{\lambda }_{2}+\delta \right)\left({\omega }_{1,2}+{\omega }_{2,2}\right)}\left\{{\int }_{0}^{u}\left({\text{e}}^{-{\omega }_{1,2}\left(u-v\right)}-{\text{e}}^{-{\omega }_{1,2}u-{\omega }_{2,2}v}\right){\gamma }_{1}\left(v\right)\text{d}v\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{\int }_{u}^{\infty }\left({\text{e}}^{{\omega }_{2,2}\left(u-v\right)}-{\text{e}}^{-{\omega }_{1,2}u-{\omega }_{2,2}v}\right){\gamma }_{2}\left(v\right)\text{d}v\right\}\end{array}$

$\begin{array}{l}=\frac{{\lambda }_{2}{\eta }_{1,2}{\eta }_{2,2}}{\left({\lambda }_{2}+\delta \right)\left({\eta }_{1,2}+{\eta }_{2,2}\right)}\left\{{\int }_{0}^{u}{\text{e}}^{-{\eta }_{1,2}\left(u-v\right)}{\gamma }_{1}\left(v\right)\text{d}v\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{u}^{\infty }{\text{e}}^{-{\eta }_{2,2}\left(u-v\right)}{\gamma }_{1}\left(v\right)\text{d}v-{\int }_{0}^{\infty }{\text{e}}^{-{\eta }_{1,2}u-{\eta }_{2,2}v}{\gamma }_{1}\left(v\right)\text{d}v\right\}\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{2{\lambda }_{2}\theta {\omega }_{1,2}{\omega }_{2,2}}{\left(2{\lambda }_{2}+\delta \right)\left({\omega }_{1,2}+{\omega }_{2,2}\right)}\left\{{\int }_{0}^{u}{\text{e}}^{-{\omega }_{1,2}\left(u-v\right)}{\gamma }_{1}\left(v\right)\text{d}v\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{u}^{\infty }{\text{e}}^{{\omega }_{2,2}\left(u-v\right)}{\gamma }_{2}\left(v\right)\text{d}v-{\int }_{0}^{\infty }{\text{e}}^{-{\omega }_{1,2}u-{\omega }_{2,2}v}{\gamma }_{2}\left(v\right)\right\}\end{array}$ (4.18)

So

$\begin{array}{c}{\varphi }_{1,w}\left(u\right)=\frac{{\lambda }_{1}{\eta }_{1,1}{\eta }_{2,1}}{\left({\lambda }_{1}+\delta \right)\left({\eta }_{1,1}+{\eta }_{2,1}\right)}\left({m}_{1,1}\left(u\right)+{T}_{{\eta }_{2,1}}{\gamma }_{1}\left(v\right)-{\text{e}}^{-{\eta }_{1,1}u}{\stackrel{˜}{\gamma }}_{1}\left({\eta }_{2,1}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{2{\lambda }_{1}\theta {\omega }_{1,1}{\omega }_{2,1}}{\left(2{\lambda }_{1}+\delta \right)\left({\omega }_{1,1}+{\omega }_{2,1}\right)}\left({m}_{1,2}\left(u\right)+{T}_{{\omega }_{2,1}}{\gamma }_{2}\left(v\right)-{\text{e}}^{-{\omega }_{1,1}u}{\stackrel{˜}{\gamma }}_{2}\left({\omega }_{2,1}\right)\right)\end{array}$

$\begin{array}{c}{\varphi }_{2,w}\left(u\right)=\frac{{\lambda }_{2}{\eta }_{1,2}{\eta }_{2,2}}{\left({\lambda }_{2}+\delta \right)\left({\eta }_{1,2}+{\eta }_{2,2}\right)}\left({m}_{2,1}\left(u\right)+{T}_{{\eta }_{2,2}}{\gamma }_{1}\left(v\right)-{\text{e}}^{-{\eta }_{1,2}u}{\stackrel{˜}{\gamma }}_{1}\left({\eta }_{2,2}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{2{\lambda }_{2}\theta {\omega }_{1,2}{\omega }_{2,2}}{\left(2{\lambda }_{2}+\delta \right)\left({\omega }_{1,2}+{\omega }_{2,2}\right)}\left({m}_{2,2}\left(u\right)+{T}_{{\omega }_{2,2}}{\gamma }_{2}\left(v\right)-{\text{e}}^{-{\omega }_{1,2}u}{\stackrel{˜}{\gamma }}_{2}\left({\omega }_{2,2}\right)\right)\end{array}$

Theorem 2. ${\varphi }_{1,d}\left(u\right)$ , ${\varphi }_{2,d}\left(u\right)$ satisfies the following integro-differential equation

$\begin{array}{c}{\varphi }_{1,d}\left(u\right)={\text{e}}^{-{\eta }_{1,1}u}+\frac{{\lambda }_{1}{\eta }_{1,1}{\eta }_{2,1}}{\left({\lambda }_{1}+\delta \right)\left({\eta }_{1,1}+{\eta }_{2,1}\right)}\left[{n}_{1,1}\left(u\right)+{T}_{{\eta }_{2,1}}{\alpha }_{1}\left(u\right)-{\text{e}}^{-{\eta }_{1,1}u}{\stackrel{˜}{\alpha }}_{1}\left({\eta }_{2,1}\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{2{\lambda }_{1}\theta {\omega }_{1,1}{\omega }_{2,1}}{\left(2{\lambda }_{1}+\delta \right)\left({\omega }_{1,1}+{\omega }_{2,1}\right)}\left[{n}_{1,2}\left(u\right)+{T}_{{\omega }_{2,1}}{\alpha }_{2}\left(u\right)-{\text{e}}^{-{\omega }_{1,1}u}{\stackrel{˜}{\alpha }}_{2}\left({\omega }_{2,1}\right)\right]\end{array}$ (4.19)

$\begin{array}{c}{\varphi }_{2,d}\left(u\right)={\text{e}}^{-{\eta }_{1,2}u}+\frac{{\lambda }_{2}{\eta }_{1,2}{\eta }_{2,2}}{\left({\lambda }_{2}+\delta \right)\left({\eta }_{1,2}+{\eta }_{2,2}\right)}\left[{n}_{2,1}\left(u\right)+{T}_{{\eta }_{2,2}}{\alpha }_{1}\left(u\right)-{\text{e}}^{-{\eta }_{1,2}u}{\stackrel{˜}{\alpha }}_{1}\left({\eta }_{2,2}\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{2{\lambda }_{2}\theta {\omega }_{1,2}{\omega }_{2,2}}{\left(2{\lambda }_{2}+\delta \right)\left({\omega }_{1,2}+{\omega }_{2,2}\right)}\left[{n}_{2,2}\left(u\right)+{T}_{{\omega }_{2,2}}{\alpha }_{2}\left(u\right)-{\text{e}}^{-{\omega }_{1,2}u}{\stackrel{˜}{\alpha }}_{2}\left({\omega }_{2,2}\right)\right]\end{array}$ (4.20)

${n}_{i,1}\left(u\right)={\int }_{0}^{u}{\text{e}}^{-{\eta }_{1,i}\left(u-v\right)}{\alpha }_{1}\left(v\right)\text{d}v,i=1,2$ , ${n}_{i,2}\left(u\right)={\int }_{0}^{u}{\text{e}}^{-{\omega }_{1,i}\left(u-v\right)}{\alpha }_{2}\left(v\right)\text{d}v,i=1,2$ .

Proof. We analyze the first claim size, and according to whether or not oscillation lead to ruin, we have

$\begin{array}{c}{\varphi }_{1,d}\left(u\right)={\int }_{0}^{\infty }{\lambda }_{1}{\text{e}}^{-\left({\lambda }_{1}+\delta \right)t}{\int }_{-\infty }^{u}\mathrm{Pr}\left(\stackrel{¯}{{W}_{1}}\left(t\right){Q}_{1}\right){\varphi }_{1,d}\left(u-y-x\right)+P\left(x>{Q}_{2}\right){\varphi }_{2,d}\left(u-y-x\right)\right]\text{d}x\text{d}t\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+E\left[{\text{e}}^{-\delta {\tau }_{1}}I\left({\tau }_{1} (4.21)

$\begin{array}{c}{\varphi }_{2,d}\left(u\right)={\int }_{0}^{\infty }{\lambda }_{2}{\text{e}}^{-\left({\lambda }_{2}+\delta \right)t}{\int }_{-\infty }^{u}\mathrm{Pr}\left(\stackrel{¯}{{W}_{2}}\left(t\right){Q}_{1}\right){\varphi }_{1,d}\left(u-y-x\right)+P\left(x>{Q}_{2}\right){\varphi }_{2,d}\left(u-y-x\right)\right]\text{d}x\text{d}t\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+E\left[{\text{e}}^{-\delta {\tau }_{2}}I\left({\tau }_{2} (4.22)

Because of the independence between V and $\left\{W\left(t\right)\right\}$ , we can get the expectation for the random variable exponentially distributed ( ${\tau }_{i},i=1,2$ is the first hitting time).

$E\left[{\text{e}}^{-\delta {\tau }_{1}}I\left({\tau }_{1} (4.23)

$E\left[{\text{e}}^{-\delta {\tau }_{2}}I\left({\tau }_{2} (4.24)

Following the proofed ${p}_{i}\left(u,y,x\right)$ , we have

$\begin{array}{l}{\varphi }_{1,d}\left(u\right)={\text{e}}^{-{\eta }_{1,1}u}+{\int }_{-\infty }^{u}{\int }_{0}^{u-y}\left[L\left(x\right){\varphi }_{1,d}\left(u-y-x\right)\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\stackrel{¯}{L}\left(x\right){\varphi }_{2,d}\left(u-y-x\right)\right]{p}_{1}\left(u,y,x\right)\text{d}x\text{d}y\\ ={\text{e}}^{-{\eta }_{1,1}u}+\frac{{\lambda }_{1}{\eta }_{1,1}{\eta }_{2,1}}{\left({\lambda }_{1}+\delta \right)\left({\eta }_{1,1}+{\eta }_{2,1}\right)}\left[{\int }_{0}^{u}{\int }_{0}^{u-y}\left[L\left(x\right){\varphi }_{1,d}\left(u-y-x\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+\stackrel{¯}{L}\left(x\right){\varphi }_{2,d}\left(u-y-x\right)\right]\left({\text{e}}^{-{\eta }_{1,1}y}-{\text{e}}^{-\left({\eta }_{1,1}+{\eta }_{2,1}\right)u+{\eta }_{2,1}y}\right)\left(f\left(x\right)-\theta h\left(x\right)\right)\text{d}x\text{d}y\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{-\infty }^{0}{\int }_{0}^{u-y}\left[L\left(x\right){\varphi }_{1,d}\left(u-y-x\right)+\stackrel{¯}{L}\left(x\right){\varphi }_{2,d}\left(u-y-x\right)\right]\end{array}$

$\begin{array}{l}\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}×\left({\text{e}}^{{\eta }_{2,1}y}-{\text{e}}^{-\left({\eta }_{1,1}+{\eta }_{2,1}\right)u+{\eta }_{2,1}y}\right)\left(f\left(x\right)-\theta h\left(x\right)\right)\text{d}x\text{d}y\right]\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{2{\lambda }_{1}\theta {\omega }_{1,1}{\omega }_{2,1}}{\left(2{\lambda }_{1}+\delta \right)\left({\omega }_{1,1}+{\omega }_{2,1}\right)}\left[{\int }_{0}^{u}{\int }_{0}^{u-y}\left[L\left(x\right){\varphi }_{1,d}\left(u-y-x\right)\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\stackrel{¯}{L}\left(x\right){\varphi }_{2,d}\left(u-y-x\right)\right]\left({\text{e}}^{-{\omega }_{1,1}y}-{\text{e}}^{-\left({\omega }_{1,1}+{\omega }_{2,1}\right)u+{\omega }_{2,1}y}\right)h\left(x\right)\text{d}x\text{d}y\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{\int }_{-\infty }^{0}{\int }_{0}^{u-y}\left[L\left(x\right){\varphi }_{1,d}\left(u-y-x\right)+\stackrel{¯}{L}\left(x\right){\varphi }_{2,d}\left(u-y-x\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }×\left({\text{e}}^{{\omega }_{2,1}y}-{\text{e}}^{-\left(\omega 1+{\omega }_{2,1}\right)u+{\omega }_{2,1}y}\right)h\left(x\right)\text{d}x\text{d}y\right]\end{array}$

$\begin{array}{l}={\text{e}}^{-{\eta }_{1,1}u}+\frac{{\lambda }_{1}{\eta }_{1,1}{\eta }_{2,1}}{\left({\lambda }_{1}+\delta \right)\left({\eta }_{1,1}+{\eta }_{2,1}\right)}\left\{{\int }_{0}^{u}\left({\text{e}}^{-{\eta }_{1,1}y}-{\text{e}}^{-\left({\eta }_{1,1}+{\eta }_{2,1}\right)u+{\eta }_{2,1}y}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }×\left[{\int }_{0}^{u-y}\left({\xi }_{1}\left(x\right){\varphi }_{1,d}\left(u-y-x\right)+{\chi }_{1}\left(x\right){\varphi }_{2,d}\left(u-y-x\right)\right)\text{d}x\right]\text{d}y\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{2{\lambda }_{1}\theta {\omega }_{1,1}{\omega }_{2,1}}{\left(2{\lambda }_{1}+\delta \right)\left({\omega }_{1,1}+{\omega }_{2,1}\right)}\left\{{\int }_{0}^{u}\left({\text{e}}^{-{\omega }_{1,1}y}-{\text{e}}^{-\left({\omega }_{1,1}+{\omega }_{2,1}\right)u+{\omega }_{2,1}y}\right)\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}×{\int }_{0}^{u-y}\left({\xi }_{1}\left(x\right){\varphi }_{1,d}\left(u-y-x\right)+{\chi }_{1}\left(x\right){\varphi }_{2,d}\left(u-y-x\right)\right)\text{d}x\text{d}y\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{-\infty }^{0}\left({\text{e}}^{{\omega }_{2,1}y}-{\text{e}}^{-\left({\omega }_{1,1}+{\omega }_{2,1}\right)u+{\omega }_{2,1}y}\right){\int }_{0}^{u-y}\left[{\xi }_{2}{\varphi }_{1,d}\left(u-y-x\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{\chi }_{2}\left(x\right){\varphi }_{2,d}\left(u-y-x\right)\right]\text{d}x\text{d}y\right\}\end{array}$ (4.25)

$\begin{array}{l}{\varphi }_{2,d}\left(u\right)={\text{e}}^{-{\eta }_{1,2}u}+\frac{{\lambda }_{2}{\eta }_{1,2}{\eta }_{2,2}}{\left({\lambda }_{2}+\delta \right)\left({\eta }_{1,2}+{\eta }_{2,2}\right)}\left\{{\int }_{0}^{u}\left({\text{e}}^{-{\eta }_{1,2}y}-{\text{e}}^{-\left({\eta }_{1,2}+{\eta }_{2,2}\right)u+{\eta }_{2,2}y}\right)\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}×\left[{\int }_{0}^{u-y}\left({\xi }_{1}\left(x\right){\varphi }_{1,d}\left(u-y-x\right)+{\chi }_{1}\left(x\right){\varphi }_{2,d}\left(u-y-x\right)\right)\text{d}x\right]\text{d}y\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{-\infty }^{0}\left({\text{e}}^{{\eta }_{2,2}y}-{\text{e}}^{-\left({\eta }_{1,2}+{\eta }_{2,2}\right)u+{\eta }_{2,2}y}\right){\int }_{0}^{u-y}\left({\xi }_{1}\left(x\right){\varphi }_{1,d}\left(u-y-x\right)\right\}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{\chi }_{1}\left(x\right){\varphi }_{2,d}\left(u-y-x\right)\right)\text{d}x\text{d}y+\frac{2{\lambda }_{2}\theta {\omega }_{1,2}{\omega }_{2,2}}{\left(2{\lambda }_{2}+\delta \right)\left({\omega }_{1,2}+{\omega }_{2,2}\right)}\end{array}$

$\begin{array}{l}\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}×\left\{{\int }_{0}^{u}\left({\text{e}}^{-{\omega }_{1,2}y}-{\text{e}}^{-\left({\omega }_{1,2}+{\omega }_{2,2}\right)u+{\omega }_{2,2}y}\right){\int }_{0}^{u-y}\left({\xi }_{1}\left(x\right){\varphi }_{1,d}\left(u-y-x\right)\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\chi }_{1}\left(x\right){\varphi }_{2,d}\left(u-y-x\right)\right)\text{d}x\text{d}y+{\int }_{-\infty }^{0}\left({\text{e}}^{{\omega }_{2,2}y}-{\text{e}}^{-\left({\omega }_{1,2}+{\omega }_{2,2}\right)u+{\omega }_{2,2}y}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }×{\int }_{0}^{u-y}\left[{\xi }_{2}{\varphi }_{1,d}\left(u-y-x\right)+{\chi }_{2}\left(x\right){\varphi }_{2,d}\left(u-y-x\right)\right]\text{d}x\text{d}y\right\}\end{array}$ (4.26)

We bring the change of variable $v=u-y$ into (4.25) and (4.26)

$\begin{array}{l}{\varphi }_{1,d}={\text{e}}^{-{\eta }_{1,1}u}+\frac{{\lambda }_{1}{\eta }_{1,1}{\eta }_{2,1}}{\left({\lambda }_{1}+\delta \right)\left({\eta }_{1,1}+{\eta }_{2,1}\right)}\left\{{\int }_{0}^{u}\left({\text{e}}^{-{\eta }_{1,1}\left(u-v\right)}-{\text{e}}^{-{\eta }_{1,1}u-{\eta }_{2,1}v}\right)\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}×\left[{\int }_{0}^{v}\left({\xi }_{1}\left(x\right){\varphi }_{1,d}\left(v-x\right)+{\chi }_{1}\left(x\right){\varphi }_{2,d}\left(v-x\right)\right)\text{d}x\right]\text{d}v\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{\int }_{u}^{\infty }\left({\text{e}}^{{\eta }_{2,1}\left(u-v\right)}-{\text{e}}^{-{\eta }_{1,1}u-{\eta }_{2,1}v}\right){\int }_{0}^{v}\left({\xi }_{1}\left(x\right){\varphi }_{1,d}\left(v-x\right)+{\chi }_{1}\left(x\right){\varphi }_{2,d}\left(v-x\right)\right)\text{d}x\text{d}v\right\}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+\frac{2{\lambda }_{1}\theta {\omega }_{1,1}{\omega }_{2,1}}{\left(2{\lambda }_{1}+\delta \right)\left({\omega }_{1,1}+{\omega }_{2,1}\right)}\left\{{\int }_{0}^{u}\left({\text{e}}^{-{\omega }_{1,1}\left(u-v\right)}-{\text{e}}^{-{\omega }_{1,1}u-{\omega }_{2,1}v}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }×{\int }_{0}^{v}\left({\xi }_{2}\left(x\right){\varphi }_{1,d}\left(v-x\right)+{\chi }_{2}\left(x\right){\varphi }_{2,d}\left(v-x\right)\right)\text{d}x\text{d}v\end{array}$

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{\int }_{u}^{\infty }\left({\text{e}}^{{\omega }_{2,1}\left(u-v\right)}-{\text{e}}^{-{\omega }_{1,1}u{\omega }_{2,1}v}\right){\int }_{0}^{v}\left[{\xi }_{2}{\varphi }_{1,d}\left(v-x\right)+{\chi }_{2}\left(x\right){\varphi }_{2,d}\left(v-x\right)\right]\text{d}x\text{d}v\right\}\\ ={\text{e}}^{-{\eta }_{1,1}u}+\frac{{\lambda }_{1}{\eta }_{1,1}{\eta }_{2,1}}{\left({\lambda }_{1}+\delta \right)\left({\eta }_{1,1}+{\eta }_{2,1}\right)}\left\{{\int }_{0}^{u}\left({\text{e}}^{-{\eta }_{1,1}\left(u-v\right)}-{\text{e}}^{-{\eta }_{1,1}u-{\eta }_{2,1}v}\right){\alpha }_{1}\left(v\right)\text{d}v\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{\int }_{u}^{\infty }\left({\text{e}}^{{\eta }_{2,1}v}-{\text{e}}^{-{\eta }_{1,1}u-{\eta }_{2,1}v}\right){\alpha }_{1}\left(v\right)\text{d}v\right\}+\frac{2{\lambda }_{1}\theta {\omega }_{1,1}{\omega }_{2,1}}{\left(2{\lambda }_{1}+\delta \right)\left({\omega }_{1,1}+{\omega }_{2,1}\right)}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }×\left\{{\int }_{0}^{u}\left({\text{e}}^{-{\omega }_{1,1}\left(u-v\right)}-{\text{e}}^{-{\omega }_{1,1}u-{\omega }_{2,1}v}\right){\alpha }_{2}\left(v\right)\text{d}v+{\int }_{u}^{\infty }\left({\text{e}}^{{\omega }_{2,1}\left(u-v\right)}-{\text{e}}^{-{\omega }_{1,1}u-{\omega }_{2,1}v}\right){\alpha }_{2}\left(v\right)\text{d}v\right\}\end{array}$

$\begin{array}{l}={\text{e}}^{-{\eta }_{1,1}u}+\frac{{\lambda }_{1}{\eta }_{1,1}{\eta }_{2,1}}{\left({\lambda }_{1}+\delta \right)\left({\eta }_{1,1}+{\eta }_{2,1}\right)}\left\{{\int }_{0}^{u}{\text{e}}^{-{\eta }_{1,1}\left(u-v\right)}{\alpha }_{1}\left(v\right)\text{d}v+{\int }_{u}^{\infty }{\text{e}}^{{\eta }_{2,1}v}{\alpha }_{1}\left(v\right)\text{d}v\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}-{\int }_{0}^{\infty }{\text{e}}^{-{\eta }_{1,1}u-{\eta }_{2,1}v}{\alpha }_{1}\left(v\right)\text{d}v\right\}+\frac{2{\lambda }_{1}\theta {\omega }_{1,1}{\omega }_{2,1}}{\left(2{\lambda }_{1}+\delta \right)\left({\omega }_{1,1}+{\omega }_{2,1}\right)}\left\{{\int }_{0}^{u}{\text{e}}^{-{\omega }_{1,1}\left(u-v\right)}{\alpha }_{2}\left(v\right)\text{d}v\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{u}^{\infty }{\text{e}}^{{\omega }_{2,1}y}{\alpha }_{2}\left(v\right)\text{d}v-{\int }_{0}^{\infty }{\text{e}}^{-{\omega }_{1,1}u-{\omega }_{2,1}v}{\alpha }_{2}\left(v\right)\text{d}v\right\}\\ ={\text{e}}^{-{\eta }_{1,1}u}+\frac{{\lambda }_{1}{\eta }_{1,1}{\eta }_{2,1}}{\left({\lambda }_{1}+\delta \right)\left({\eta }_{1,1}+{\eta }_{2,1}\right)}\left[{n}_{1,1}\left(u\right)+{T}_{{\eta }_{2,1}}{\alpha }_{1}\left(u\right)-{\text{e}}^{-{\eta }_{1,1}u}{\stackrel{˜}{\alpha }}_{1}\left({\eta }_{2,1}\right)\right]\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{2{\lambda }_{1}\theta {\omega }_{1,1}{\omega }_{2,1}}{\left(2{\lambda }_{1}+\delta \right)\left({\omega }_{1,1}+{\omega }_{2,1}\right)}\left[{n}_{1,2}\left(u\right)+{T}_{{\omega }_{2,1}}{\alpha }_{2}\left(u\right)-{\text{e}}^{-{\omega }_{1,1}u}{\stackrel{˜}{\alpha }}_{2}\left({\omega }_{2,1}\right)\right]\end{array}$

$\begin{array}{l}{\varphi }_{2,d}\left(u\right)={\text{e}}^{-{\eta }_{1,2}u}+\frac{{\lambda }_{2}{\eta }_{1,2}{\eta }_{2,2}}{\left({\lambda }_{2}+\delta \right)\left({\eta }_{1,2}+{\eta }_{2,2}\right)}\left\{{\int }_{0}^{u}\left({\text{e}}^{-{\eta }_{1,2}\left(u-v\right)}-{\text{e}}^{-{\eta }_{1,2}u-{\eta }_{2,2}v}\right){\alpha }_{1}\left(v\right)\text{d}v\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{u}^{\infty }\left({\text{e}}^{{\eta }_{2,2}\left(u-v\right)}-{\text{e}}^{-{\eta }_{1,2}u-{\eta }_{2,2}v}\right){\alpha }_{1}\left(v\right)\text{d}v\right\}+\frac{2{\lambda }_{2}\theta {\omega }_{1,2}{\omega }_{2,2}}{\left(2{\lambda }_{2}+\delta \right)\left({\omega }_{1,2}+{\omega }_{2,2}\right)}\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}×\left\{{\int }_{0}^{u}\left({\text{e}}^{-{\omega }_{1,2}\left(u-v\right)}-{\text{e}}^{-{\omega }_{1,2}u-{\omega }_{2,2}v}\right){\alpha }_{2}\left(v\right)\text{d}v+{\int }_{u}^{\infty }\left({\text{e}}^{{\omega }_{2,2}\left(u-v\right)}-{\text{e}}^{-{\omega }_{1,2}u-{\omega }_{2,2}v}\right){\alpha }_{2}\left(v\right)\text{d}v\right\}\\ ={\text{e}}^{-{\eta }_{1,2}u}+\frac{{\lambda }_{2}{\eta }_{1,2}{\eta }_{2,2}}{\left({\lambda }_{2}+\delta \right)\left({\eta }_{1,2}+{\eta }_{2,2}\right)}\left\{{\int }_{0}^{u}{\text{e}}^{-{\eta }_{1,2}\left(u-v\right)}{\alpha }_{1}\left(v\right)\text{d}v+{\int }_{u}^{\infty }{\text{e}}^{{\eta }_{2,2}\left(u-v\right)}{\alpha }_{1}\left(v\right)\text{d}v\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}-{\int }_{0}^{\infty }{\text{e}}^{-{\eta }_{1,2}u-{\eta }_{2,2}v}{\alpha }_{1}\left(v\right)\text{d}v+\frac{2{\lambda }_{2}\theta {\omega }_{1,2}{\omega }_{2,2}}{\left(2{\lambda }_{2}+\delta \right)\left({\omega }_{1,2}+{\omega }_{2,2}\right)}\left\{{\int }_{0}^{u}{\text{e}}^{-{\omega }_{1,2}\left(u-v\right)}{\alpha }_{2}\left(v\right)\text{d}v\end{array}$

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{u}^{\infty }{\text{e}}^{{\omega }_{2,2}\left(u-v\right)}{\alpha }_{2}\left(v\right)\text{d}v-{\int }_{0}^{\infty }{\text{e}}^{-{\omega }_{1,2}u-{\omega }_{2,2}v}{\alpha }_{2}\left(v\right)\text{d}v\right\}\\ ={\text{e}}^{-{\eta }_{1,2}u}+\frac{{\lambda }_{2}{\eta }_{1,2}{\eta }_{2,2}}{\left({\lambda }_{2}+\delta \right)\left({\eta }_{1,2}+{\eta }_{2,2}\right)}\left[{n}_{2,1}\left(u\right)+{T}_{{\eta }_{2,2}}{\alpha }_{1}\left(u\right)-{\text{e}}^{-{\eta }_{1,2}u}{\stackrel{˜}{\alpha }}_{1}\left({\eta }_{2,2}\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+\frac{2{\lambda }_{2}\theta {\omega }_{1,2}{\omega }_{2,2}}{\left(2{\lambda }_{2}+\delta \right)\left({\omega }_{1,2}+{\omega }_{2,2}\right)}\left[{n}_{2,2}\left(u\right)+{T}_{{\omega }_{2,2}}{\alpha }_{2}\left(u\right)-{\text{e}}^{-{\omega }_{1,2}u}{\stackrel{˜}{\alpha }}_{2}\left({\omega }_{2,2}\right)\right]\end{array}$

and

${\alpha }_{1}\left(y\right)={\int }_{0}^{y}\left[{\xi }_{1}{\varphi }_{1,d}\left(y-x\right)+{\chi }_{1}\left(x\right){\varphi }_{2,d}\left(y-x\right)\right]\text{d}x$ ,

${\alpha }_{2}\left(y\right)={\int }_{0}^{y}\left[{\xi }_{2}{\varphi }_{1,d}\left(y-x\right)+{\chi }_{2}\left(x\right){\varphi }_{2,d}\left(y-x\right)\right]\text{d}x$ .

5. Laplace Transforms

Firstly, we consider the Laplace transforms of ${m}_{1,i}\left(u\right)$

$\begin{array}{c}{\stackrel{˜}{m}}_{1,1}\left(s\right)={\int }_{0}^{\infty }{\text{e}}^{-sy}{\int }_{0}^{y}{\text{e}}^{-{\eta }_{1,1}\left(y-v\right)}{\gamma }_{1}\left(v\right)\text{d}v\text{d}y={\int }_{0}^{\infty }{\int }_{v}^{\infty }{\text{e}}^{-sy}{\text{e}}^{-{\eta }_{1,1}\left(y-v\right)}{\gamma }_{1}\left(v\right)\text{d}v\text{d}y\\ ={\int }_{0}^{\infty }{\text{e}}^{{\eta }_{1,1}v}\frac{1}{s+{\eta }_{1,1}}{\text{e}}^{-\left(s+{\eta }_{1,1}\right)v}{\gamma }_{1}\left(v\right)\text{d}v\\ =\frac{1}{s+{\eta }_{1,1}}{\int }_{0}^{\infty }{\text{e}}^{-sv}{\gamma }_{1}\left(v\right)\text{d}v=\frac{1}{s+{\eta }_{1,1}}{\stackrel{˜}{\gamma }}_{1}\left(v\right)\end{array}$ (5.1)

${\int }_{0}^{\infty }{\text{e}}^{-su}{\text{e}}^{-{\eta }_{1,1}v}{\stackrel{˜}{\gamma }}_{1}\left({\eta }_{2,1}\right)\text{d}u={\stackrel{˜}{\gamma }}_{1}\left({\eta }_{2,1}\right){\int }_{0}^{\infty }{\text{e}}^{-\left(s+{\eta }_{1,1}\right)u}\text{d}u=\frac{1}{s+{\eta }_{1,1}}{\stackrel{˜}{\gamma }}_{1}\left({\eta }_{2,1}\right)$ (5.2)

So

${\stackrel{˜}{m}}_{1,2}\left(s\right)={\int }_{0}^{\infty }{\text{e}}^{-sy}{\int }_{0}^{y}{\text{e}}^{-{\omega }_{1,1}\left(y-v\right)}{\gamma }_{2}\left(v\right)\text{d}v\text{d}y=\frac{1}{s+{\omega }_{1,1}}{\stackrel{˜}{\gamma }}_{2}\left(v\right)$ (5.3)

${\int }_{0}^{\infty }{\text{e}}^{-su}{\text{e}}^{-{\omega }_{1,1}v}{\stackrel{˜}{\gamma }}_{2}\left({\omega }_{2,1}\right)\text{d}u={\stackrel{˜}{\gamma }}_{2}\left({\omega }_{2,1}\right){\int }_{0}^{\infty }{\text{e}}^{-\left(s+{\omega }_{1,1}\right)u}\text{d}u=\frac{1}{s+{\omega }_{1,1}}{\stackrel{˜}{\gamma }}_{2}\left({\omega }_{2,1}\right)$ (5.4)

${\stackrel{˜}{m}}_{2,1}\left(s\right)={\int }_{0}^{\infty }{\text{e}}^{-sy}{\int }_{0}^{y}{\text{e}}^{-{\eta }_{1,2}\left(y-v\right)}{\gamma }_{1}\left(v\right)\text{d}v\text{d}y=\frac{1}{s+{\eta }_{1,2}}{\stackrel{˜}{\gamma }}_{1}\left(v\right)$ (5.5)

${\int }_{0}^{\infty }{\text{e}}^{-su}{\text{e}}^{-{\eta }_{1,2}v}{\stackrel{˜}{\gamma }}_{1}\left({\eta }_{2,2}\right)\text{d}u={\stackrel{˜}{\gamma }}_{1}\left({\eta }_{2,2}\right){\int }_{0}^{\infty }{\text{e}}^{-\left(s+{\eta }_{1,2}\right)u}\text{d}u=\frac{1}{s+{\eta }_{1,2}}{\stackrel{˜}{\gamma }}_{1}\left({\eta }_{2,2}\right)$ (5.6)

${\stackrel{˜}{m}}_{2,2}\left(s\right)={\int }_{0}^{\infty }{\text{e}}^{-sy}{\int }_{0}^{y}{\text{e}}^{-{\omega }_{1,2}\left(y-v\right)}{\gamma }_{2}\left(v\right)\text{d}v\text{d}y=\frac{1}{s+{\omega }_{1,2}}{\stackrel{˜}{\gamma }}_{2}\left(v\right)$ (5.7)

${\int }_{0}^{\infty }{\text{e}}^{-su}{\text{e}}^{-{\omega }_{1,2}v}{\stackrel{˜}{\gamma }}_{2}\left({\omega }_{2,2}\right)\text{d}u={\stackrel{˜}{\gamma }}_{2}\left({\omega }_{2,2}\right){\int }_{0}^{\infty }{\text{e}}^{-\left(s+{\omega }_{1,2}\right)u}\text{d}u=\frac{1}{s+{\omega }_{1,2}}{\stackrel{˜}{\gamma }}_{2}\left({\omega }_{2,2}\right)$ (5.8)

And then we consider the Laplace transport

$\begin{array}{l}{\stackrel{˜}{\varphi }}_{1,w}\left(s\right)=\frac{{\lambda }_{1}{\eta }_{1,1}{\eta }_{2,1}}{\left({\lambda }_{1}+\delta \right)\left({\eta }_{1,1}+{\eta }_{2,1}\right)}\left({\stackrel{˜}{m}}_{1,1}\left(s\right)+{T}_{s}{T}_{{\eta }_{2,1}}{\gamma }_{1}\left(0\right)-\frac{1}{s+{\eta }_{1,1}}{\stackrel{˜}{\gamma }}_{1}\left({\eta }_{2,1}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{2{\lambda }_{1}\theta {\omega }_{1,1}{\omega }_{2,1}}{\left(2{\lambda }_{1}+\delta \right)\left({\omega }_{1,1}+{\omega }_{2,1}\right)}\left({\stackrel{˜}{m}}_{1,2}\left(s\right)+{T}_{s}{T}_{{\omega }_{2,1}}{\gamma }_{2}\left(0\right)-\frac{1}{s+{\omega }_{1,1}}{\stackrel{˜}{\gamma }}_{2}\left({\omega }_{2,1}\right)\right)\\ =\frac{{\lambda }_{1}{\eta }_{1,1}{\eta }_{2,1}}{\left({\lambda }_{1}+\delta \right)\left({\eta }_{1,1}+{\eta }_{2,1}\right)}\left(\frac{{\stackrel{˜}{\gamma }}_{1}\left(s\right)-{\stackrel{˜}{\gamma }}_{1}\left({\eta }_{2,1}\right)}{s+{\eta }_{1,1}}-\frac{{\stackrel{˜}{\gamma }}_{1}\left(s\right)-{\stackrel{˜}{\gamma }}_{1}\left({\eta }_{2,1}\right)}{s-{\eta }_{2,1}}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{2{\lambda }_{1}\theta {\omega }_{1,1}{\omega }_{2,1}}{\left(2{\lambda }_{1}+\delta \right)\left({\omega }_{1,1}+{\omega }_{2,1}\right)}\left(\frac{{\stackrel{˜}{\gamma }}_{2}\left(s\right)-{\stackrel{˜}{\gamma }}_{2}\left({\omega }_{2,1}\right)}{s+{\omega }_{1,1}}-\frac{{\stackrel{˜}{\gamma }}_{2}\left(s\right)-{\stackrel{˜}{\gamma }}_{2}\left({\omega }_{2,1}\right)}{s-{\omega }_{2,1}}\right)\\ =\frac{{\lambda }_{1}{\eta }_{1,1}{\eta }_{2,1}}{\left({\lambda }_{1}+\delta \right)\left(s+{\eta }_{1,1}\right)\left(s-{\eta }_{2,1}\right)}\left({\stackrel{˜}{\gamma }}_{1}\left({\eta }_{2,1}\right)-{\stackrel{˜}{\gamma }}_{1}\left(s\right)\right)\end{array}$

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{2{\lambda }_{1}\theta {\omega }_{1,1}{\omega }_{2,1}}{\left(2{\lambda }_{1}+\delta \right)\left(s+{\omega }_{1,1}\right)\left(s-{\omega }_{2,1}\right)}\left({\stackrel{˜}{\gamma }}_{2}\left({\omega }_{2,1}\right)-{\stackrel{˜}{\gamma }}_{2}\left(s\right)\right)\\ =\frac{{\lambda }_{1}{\eta }_{1,1}{\eta }_{2,1}}{\left({\lambda }_{1}+\delta \right)\left(s+{\eta }_{1,1}\right)\left(s-{\eta }_{2,1}\right)}\left({\stackrel{˜}{\xi }}_{1}\left({\eta }_{2,1}\right){\stackrel{˜}{\varphi }}_{1,w}\left({\eta }_{2,1}\right)+{\stackrel{˜}{\chi }}_{1}\left({\eta }_{2,1}\right){\stackrel{˜}{\varphi }}_{2,w}\left({\eta }_{2,1}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\stackrel{˜}{\phi }}_{f-\theta h}\left({\eta }_{2,1}\right)-{\stackrel{˜}{\xi }}_{1}\left(s\right){\stackrel{˜}{\varphi }}_{1,w}\left(s\right)-{\stackrel{˜}{\chi }}_{1}\left(s\right){\stackrel{˜}{\varphi }}_{2,w}\left(s\right)-{\stackrel{˜}{\phi }}_{f-\theta h}\left(s\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{2{\lambda }_{1}\theta {\omega }_{1,1}{\omega }_{2,1}}{\left(2{\lambda }_{1}+\delta \right)\left(s+{\omega }_{1,1}\right)\left(s-{\omega }_{2,1}\right)}\left({\stackrel{˜}{\xi }}_{2}\left({\omega }_{2,1}\right){\stackrel{˜}{\varphi }}_{1,w}\left({\omega }_{2,1}\right)+{\stackrel{˜}{\chi }}_{2}\left({\omega }_{2,1}\right){\stackrel{˜}{\varphi }}_{2,w}\left({\omega }_{2,1}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\stackrel{˜}{\phi }}_{h}\left({\omega }_{2,1}\right)-{\stackrel{˜}{\xi }}_{2}\left(s\right){\stackrel{˜}{\varphi }}_{1,w}\left(s\right)-{\stackrel{˜}{\chi }}_{2}\left(s\right){\stackrel{˜}{\varphi }}_{2,w}\left(s\right)-{\stackrel{˜}{\phi }}_{h}\left(s\right)\right)\end{array}$ (5.9)

$\begin{array}{l}{\stackrel{˜}{\varphi }}_{2,w}\left(s\right)=\frac{{\lambda }_{2}{\eta }_{1,2}{\eta }_{2,2}}{\left({\lambda }_{2}+\delta \right)\left({\eta }_{1,2}+{\eta }_{2,2}\right)}\left({\stackrel{˜}{m}}_{2,1}\left(s\right)+{T}_{s}{T}_{{\eta }_{2,2}}{\gamma }_{1}\left(0\right)-\frac{1}{s+{\eta }_{1,2}}{\stackrel{˜}{\gamma }}_{1}\left({\eta }_{2,2}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{2{\lambda }_{2}\theta {\omega }_{1,2}{\omega }_{2,2}}{\left(2{\lambda }_{2}+\delta \right)\left({\omega }_{1,2}+{\omega }_{2,2}\right)}\left({\stackrel{˜}{m}}_{2,2}\left(s\right)+{T}_{s}{T}_{{\omega }_{2,2}}{\gamma }_{2}\left(0\right)-\frac{1}{s+{\omega }_{1,2}}{\stackrel{˜}{\gamma }}_{2}\left({\omega }_{2,2}\right)\right)\\ =\frac{{\lambda }_{2}{\eta }_{1,2}{\eta }_{2,2}}{\left({\lambda }_{2}+\delta \right)\left({\eta }_{1,2}+{\eta }_{2,2}\right)}\left(\frac{{\stackrel{˜}{\gamma }}_{1}\left(s\right)-{\stackrel{˜}{\gamma }}_{1}\left({\eta }_{2,2}\right)}{s+{\eta }_{1,2}}-\frac{{\stackrel{˜}{\gamma }}_{1}\left(s\right)-{\stackrel{˜}{\gamma }}_{1}\left({\eta }_{2,2}\right)}{s-{\eta }_{2,2}}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{2{\lambda }_{2}\theta {\omega }_{1,2}{\omega }_{2,2}}{\left(2{\lambda }_{2}+\delta \right)\left({\omega }_{1,2}+{\omega }_{2,2}\right)}\left(\frac{{\stackrel{˜}{\gamma }}_{2}\left(s\right)-{\stackrel{˜}{\gamma }}_{2}\left({\omega }_{2,2}\right)}{s+{\omega }_{1,2}}-\frac{{\stackrel{˜}{\gamma }}_{2}\left({\omega }_{2,2}\right)-{\stackrel{˜}{\gamma }}_{2}\left({\omega }_{2,2}\right)}{s-{\omega }_{2,2}}\right)\\ =\frac{{\lambda }_{2}{\eta }_{1,2}{\eta }_{2,2}}{\left({\lambda }_{2}+\delta \right)\left(s+{\eta }_{1,2}\right)\left(s-{\eta }_{2,2}\right)}\left({\stackrel{˜}{\gamma }}_{1}\left({\eta }_{2,2}\right)-{\stackrel{˜}{\gamma }}_{1}\left(s\right)\right)\end{array}$

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{2{\lambda }_{2}\theta {\omega }_{1,2}{\omega }_{2,2}}{\left(2{\lambda }_{2}+\delta \right)\left(s+{\omega }_{1,2}\right)\left(s-{\omega }_{2,2}\right)}\left({\stackrel{˜}{\gamma }}_{2}\left({\omega }_{2,2}\right)-{\stackrel{˜}{\gamma }}_{2}\left(s\right)\right)\\ =\frac{{\lambda }_{1}{\eta }_{1,1}{\eta }_{2,1}}{\left({\lambda }_{1}+\delta \right)\left(s+{\eta }_{1,1}\right)\left(s-{\eta }_{2,1}\right)}\left({\stackrel{˜}{\xi }}_{1}\left({\eta }_{2,2}\right){\stackrel{˜}{\varphi }}_{1,w}\left({\eta }_{2,2}\right)+{\stackrel{˜}{\chi }}_{1}\left({\eta }_{2,2}\right){\stackrel{˜}{\varphi }}_{2,w}\left({\eta }_{2,2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\stackrel{˜}{\phi }}_{f-\theta h}\left({\eta }_{2,2}\right)-{\stackrel{˜}{\xi }}_{1}\left(s\right){\stackrel{˜}{\varphi }}_{1,w}\left(s\right)-{\stackrel{˜}{\chi }}_{1}\left(s\right){\stackrel{˜}{\varphi }}_{2,w}\left(s\right)-{\stackrel{˜}{\phi }}_{f-\theta h}\left(s\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{2{\lambda }_{1}\theta {\omega }_{1,1}{\omega }_{2,1}}{\left(2{\lambda }_{1}+\delta \right)\left(s+{\omega }_{1,1}\right)\left(s-{\omega }_{2,1}\right)}\left({\stackrel{˜}{\xi }}_{2}\left({\omega }_{2,2}\right){\stackrel{˜}{\varphi }}_{1,w}\left({\omega }_{2,2}\right)+{\stackrel{˜}{\chi }}_{2}\left({\omega }_{2,2}\right){\stackrel{˜}{\varphi }}_{2,w}\left({\omega }_{2,2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\stackrel{˜}{\phi }}_{h}\left({\omega }_{2,2}\right)-{\stackrel{˜}{\xi }}_{2}\left(s\right){\stackrel{˜}{\varphi }}_{1,w}\left(s\right)-{\stackrel{˜}{\chi }}_{2}\left(s\right){\stackrel{˜}{\varphi }}_{2,w}\left(s\right)-{\stackrel{˜}{\phi }}_{h}\left(s\right)\right)\end{array}$ (5.10)

And

${\stackrel{˜}{\varphi }}_{1,w}\left({\eta }_{2,1}\right)=\frac{2{\lambda }_{1}\theta {\omega }_{1,1}{\omega }_{2,1}}{\left(2{\lambda }_{1}+\delta \right)\left({\eta }_{2,1}+{\omega }_{1,1}\right)\left({\eta }_{2,1}-{\omega }_{2,1}\right)}\left({\stackrel{˜}{\gamma }}_{2}\left({\omega }_{2,1}\right)-{\stackrel{˜}{\gamma }}_{2}\left({\eta }_{2,1}\right)\right)$ .

${\stackrel{˜}{\varphi }}_{1,w}\left({\omega }_{2,1}\right)=\frac{{\lambda }_{1}{\eta }_{1,1}{\eta }_{2,1}}{\left({\lambda }_{1}+\delta \right)\left({\omega }_{2,1}+{\eta }_{1,1}\right)\left({\omega }_{2,1}-{\eta }_{2,1}\right)}\left({\stackrel{˜}{\gamma }}_{1}\left({\eta }_{2,1}\right)-{\stackrel{˜}{\gamma }}_{1}\left({\omega }_{2,1}\right)\right)$ .

${\stackrel{˜}{\varphi }}_{2,w}\left({\eta }_{2,2}\right)=\frac{2{\lambda }_{2}\theta {\omega }_{1,2}{\omega }_{2,2}}{\left(2{\lambda }_{2}+\delta \right)\left({\eta }_{2,2}+{\omega }_{1,2}\right)\left({\eta }_{2,2}-{\omega }_{2,2}\right)}\left({\stackrel{˜}{\gamma }}_{2}\left({\omega }_{2,2}\right)-{\stackrel{˜}{\gamma }}_{2}\left({\eta }_{2,2}\right)\right)$ .

${\stackrel{˜}{\varphi }}_{2,w}\left({\omega }_{2,2}\right)=\frac{{\lambda }_{2}{\eta }_{1,2}{\eta }_{2,2}}{\left({\lambda }_{2}+\delta \right)\left({\omega }_{2,2}+{\eta }_{1,2}\right)\left(s-{\eta }_{2,2}\right)}\left({\stackrel{˜}{\gamma }}_{1}\left({\eta }_{2,2}\right)-{\stackrel{˜}{\gamma }}_{1}\left({\omega }_{2,2}\right)\right)$ .

Theorem 3. The Laplace transforms of the Gerber-Shiu functions given (4.8) and (4.9) are

$\begin{array}{l}\left(1+{A}_{1}{\stackrel{˜}{\xi }}_{1}\left(s\right)+{B}_{1}{\stackrel{˜}{\xi }}_{2}\left(s\right)\right){\stackrel{˜}{\varphi }}_{1,w}\left(s\right)+\left({A}_{1}{\stackrel{˜}{\chi }}_{1}\left(s\right)+{B}_{1}{\stackrel{˜}{\chi }}_{2}\left(s\right)\right){\stackrel{˜}{\varphi }}_{2,w}\left(s\right)\\ ={A}_{1}\left({\stackrel{˜}{\xi }}_{1}\left({\eta }_{2,1}\right){\stackrel{˜}{\varphi }}_{1,w}\left({\eta }_{2,1}\right)+{\stackrel{˜}{\chi }}_{1}\left({\eta }_{2,1}\right){\stackrel{˜}{\varphi }}_{2,w}\left({\eta }_{2,1}\right)+{\stackrel{˜}{\phi }}_{f-\theta h}\left({\eta }_{2,1}\right)-{\stackrel{˜}{\phi }}_{f-\theta h}\left(s\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{B}_{1}\left({\stackrel{˜}{\xi }}_{2}\left({\omega }_{2,1}\right){\stackrel{˜}{\varphi }}_{1,w}\left({\omega }_{2,1}\right)+{\stackrel{˜}{\chi }}_{2}\left({\omega }_{2,1}\right){\stackrel{˜}{\varphi }}_{2,w}\left({\omega }_{2,1}\right)+{\stackrel{˜}{\phi }}_{h}\left({\omega }_{2,1}\right)-{\stackrel{˜}{\phi }}_{h}\left(s\right)\right)\end{array}$ (4.11)

$\begin{array}{l}\left(1+{A}_{2}{\stackrel{˜}{\xi }}_{1}\left(s\right)+{B}_{2}{\stackrel{˜}{\xi }}_{2}\left(s\right)\right){\stackrel{˜}{\varphi }}_{1,w}\left(s\right)+\left({A}_{2}{\stackrel{˜}{\chi }}_{1}\left(s\right)+{B}_{2}{\stackrel{˜}{\chi }}_{2}\left(s\right)\right){\stackrel{˜}{\varphi }}_{2,w}\left(s\right)-{\stackrel{˜}{\phi }}_{f-\theta h}\left(s\right)\right)\\ ={A}_{2}\left({\stackrel{˜}{\xi }}_{1}\left({\eta }_{2,2}\right){\stackrel{˜}{\varphi }}_{1,w}\left({\eta }_{2,2}\right)+{\stackrel{˜}{\chi }}_{1}\left({\eta }_{2,2}\right){\stackrel{˜}{\varphi }}_{2,w}\left({\eta }_{2,2}\right)+{\stackrel{˜}{\phi }}_{f-\theta h}\left({\eta }_{2,2}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{B}_{2}\left({\stackrel{˜}{\xi }}_{2}\left({\omega }_{2,2}\right){\stackrel{˜}{\varphi }}_{1,w}\left({\omega }_{2,2}\right)+{\stackrel{˜}{\chi }}_{2}\left({\omega }_{2,2}\right){\stackrel{˜}{\varphi }}_{2,w}\left({\omega }_{2,2}\right)+{\stackrel{˜}{\phi }}_{h}\left({\omega }_{2,2}\right)-{\stackrel{˜}{\phi }}_{h}\left(s\right)\right)\end{array}$ (4.12)

${A}_{1}=\frac{{\lambda }_{1}{\eta }_{1,1}{\eta }_{2,1}}{\left({\lambda }_{1}+\delta \right)\left(s+{\eta }_{1,1}\right)\left(s-{\eta }_{2,1}\right)}$ , ${B}_{1}=\frac{2{\lambda }_{1}\theta {\omega }_{1,1}{\omega }_{2,1}}{\left(2{\lambda }_{1}+\delta \right)\left(s+{\omega }_{1,1}\right)\left(s-{\omega }_{2,1}\right)}$ ,

${A}_{2}=\frac{{\lambda }_{2}{\eta }_{1,2}{\eta }_{2,2}}{\left({\lambda }_{2}+\delta \right)\left(s+{\eta }_{1,2}\right)\left(s-{\eta }_{2,2}\right)}$ , ${B}_{2}=\frac{2{\lambda }_{1}\theta {\omega }_{1,2}{\omega }_{2,2}}{\left(2{\lambda }_{1}+\delta \right)\left(s+{\omega }_{1,2}\right)\left(s-{\omega }_{2,2}\right)}$ .

And then, we consider the Laplace transforms of ${\stackrel{˜}{n}}_{1,i}\left(s\right)$

${\stackrel{˜}{n}}_{1,1}\left(s\right)={\int }_{0}^{\infty }{\text{e}}^{-sy}{\int }_{0}^{y}{\text{e}}^{-{\eta }_{1,1}\left(y-v\right)}{\alpha }_{1}\left(v\right)\text{d}v\text{d}y=\frac{1}{s+{\eta }_{1,1}}{\stackrel{˜}{\alpha }}_{1}\left(s\right)$ (5.13)

${\int }_{0}^{\infty }{\text{e}}^{-su}{\text{e}}^{-{\eta }_{1,1}u}{\stackrel{˜}{\alpha }}_{1}\left({\eta }_{2,1}\right)\text{d}u={\stackrel{˜}{\alpha }}_{1}\left({\eta }_{2,1}\right){\int }_{0}^{\infty }{\text{e}}^{-\left(s+{\eta }_{1,1}\right)u}\text{d}u=\frac{1}{s+{\eta }_{1,1}}{\stackrel{˜}{\alpha }}_{1}\left({\eta }_{2,1}\right)$ (5.14)

${\stackrel{˜}{n}}_{1,2}\left(s\right)={\int }_{0}^{\infty }{\text{e}}^{-sy}{\int }_{0}^{y}{\text{e}}^{-{\omega }_{1,1}\left(y-v\right)}{\alpha }_{2}\left(v\right)\text{d}v=\frac{1}{s+{\omega }_{1,1}}{\stackrel{˜}{\alpha }}_{2}\left(s\right)$ (5.15)

${\int }_{0}^{\infty }{\text{e}}^{-su}{\text{e}}^{-{\omega }_{1,1}u}{\stackrel{˜}{\alpha }}_{2}\left({\omega }_{2,1}\right)\text{d}u={\stackrel{˜}{\alpha }}_{2}\left({\omega }_{2,1}\right){\int }_{0}^{\infty }{\text{e}}^{-\left(s+{\omega }_{1,1}\right)u}\text{d}u=\frac{1}{s+{\omega }_{1,1}}{\stackrel{˜}{\alpha }}_{2}\left({\omega }_{2,1}\right)$ (5.16)

${\stackrel{˜}{n}}_{2,1}\left(s\right)={\int }_{0}^{\infty }{\text{e}}^{-sy}{\int }_{0}^{y}{\text{e}}^{-{\eta }_{1,2}\left(y-v\right)}{\alpha }_{1}\left(v\right)\text{d}v\text{d}y=\frac{1}{s+{\eta }_{1,2}}{\stackrel{˜}{\alpha }}_{1}\left(s\right)$ (5.17)

${\int }_{0}^{\infty }{\text{e}}^{-su}{\text{e}}^{-{\eta }_{1,2}u}{\stackrel{˜}{\alpha }}_{1}\left({\eta }_{2,2}\right)\text{d}u={\stackrel{˜}{\alpha }}_{1}\left({\eta }_{2,2}\right){\int }_{0}^{\infty }{\text{e}}^{-\left(s+{\eta }_{1,2}\right)u}\text{d}u=\frac{1}{s+{\eta }_{1,2}}{\stackrel{˜}{\alpha }}_{1}\left({\eta }_{2,2}\right)$ (5.18)

${\stackrel{˜}{n}}_{2,2}\left(s\right)={\int }_{0}^{\infty }{\text{e}}^{-sy}{\int }_{0}^{y}{\text{e}}^{-{\omega }_{1}}$