一类带扰动和分红策略的相依风险模型

A Class of Dependent Risk Models with Disturbance and Dividend Strategy

摘要: 本文研究受布朗运动扰动和阈值分红策略影响的相依风险模型，即盈余低于阈值时不分红，超过阈值时以恒定速率连续支付分红，同时在模型中引入索赔阈值，规定实际索赔支付为索赔额和索赔阈值的小者。本文以Gerber-Shiu惩罚函数为研究对象，运用全概率公式、泰勒展开、布朗运动性质及极限转化等方法，推导了该风险模型的Gerber-Shiu惩罚函数满足的积分–微分方程。研究结果丰富了相依风险模型的理论体系，为保险业务的风险精确度量、分红策略优化设计提供了重要理论支撑，进一步拓展了相关模型在保险实务中的应用场景。

Abstract: This paper studies a dependent risk model influenced by Brownian motion perturbations and a threshold dividend strategy, where dividends are not paid when surplus is below the threshold and continuously paid at a constant rate when surplus exceeds the threshold. Additionally, a claim threshold is introduced into the model, stipulating that the actual claim payment is the lesser of the claim amount and the claim threshold. This paper focuses on the Gerber-Shiu penalty function and employs methods such as the total probability formula, Taylor expansion, properties of Brownian motion, and limit transformation to derive the integral-differential equations of the Gerber-Shiu penalty function under both dividend-paying and non-dividend-paying conditions. The research results enrich the theoretical system of dependent risk models, provide important theoretical support for precise risk measurement and optimal design of dividend strategies in insurance business, and further expand the application scenarios of related models in insurance practice.

1. 引言

保险风险模型中的分红策略最早由De Finetti提出，之后众多学者对分红策略展开了研究，在保险风险理论及相关领域得到广泛应用(参见文献[1]-[5])。Lin和Pavlova研究了带有阈值分红策略下的经典复合泊松模型，并推导出了该模型的Gerber-Shiu惩罚函数满足的积分–微分方程和方程对应的边界条件[6]，Wan将随机扰动与阈值分红策略相结合，推导了破产前期望折现红利支付的两类积分–微分方程和Gerber-Shiu惩罚函数的解析表达式[7]，Li和Sendova考虑带有固定阈值和随机扰动的相依模型，得到了期望贴现惩罚函数满足的积分–微分方程和Gerber-Shiu惩罚函数满足的瑕疵更新方程[8]，Sun和Zhang进一步研究了具有两类索赔和阈值分红策略的扰动风险模型，推导了Gerber-Shiu惩罚函数满足的矩阵形式的积分–微分方程及其显式解[9]。

本文研究基于文献[7]-[8]中受布朗运动扰动的相依风险模型，将固定阈值拓展为随机阈值并引入阈值分红策略(即盈余低于阈值时不分红、超过阈值时以恒定速率分红)，然后结合Albrecher提出的在再保险应用中的相依模型，即实际索赔支付为索赔额和索赔阈值的小者[10]，得到了具有阈值分红策略、索赔阈值和随机扰动的相依模型，最后参考文献[9]的研究过程，对于新的风险模型，我们得到了Gerber-Shiu惩罚函数满足的积分–微分方程。

2. 模型介绍

本文考虑以下盈余过程受布朗运动扰动的风险模型

$U\left(t\right)=u+c_{1}t-S\left(t\right)+\sigma W\left(t\right)$ $t\ge 0$ (2.1)

其中$u\ge 0$ 表示初始盈余，$c_1>0$ 为保险费率，$\left\{W\left(t\right),t\ge 0\right\}$ 是标准布朗运动，$\sigma >0$ 是一个常数，$S\left(t\right)={\displaystyle \sum _{j=1}^{N\left(t\right)}\min \left(X_i,T_j\right)}$ 表示到达时间t的总索赔金额，其中$N\left(t\right)$ 表示到时刻$t$ 的索赔次数，$T_i$ 为独立同分布阈值变量，其累积分布函数为$F(\cdot )$ ，概率密度函数为$f(\cdot )$ ，索赔金额$X_1,X_2,\cdots$ 是独立同分布的随机变量，其累积分布函数为$B(\cdot )$ ，概率密度函数为$b(\cdot )$ ，均值为$\mu$ ，索赔间隔时间$\left\{W_i,i\ge 1\right\}$ 是独立同分布的随机变量序列，若索赔金额$X_i$ 大于某个随机阈值$T_i$ ，则保险公司的赔付金额为$T_i$ ，且到下一次索赔的时间服从参数为${\lambda }_1$ 的指数分布，此时分布函数为$F(\cdot )$ ，盈余过程被归为第1类；若$X_i$ 小于$T_i$ ，保险公司的赔付金额为$X_i$ ，且到下一次索赔的时间服从参数为${\lambda }_2$ 的指数分布，此时分布函数为$B(\cdot )$ ，盈余过程被归为第2类，当盈余低于阈值b时不分红，超过b时，以恒定速率d连续支付分红，此时的保费率为$c_2=c_1-d$ 。

本文定义Gerber-Shiu函数为

$V_k\left(u\right)=E\left[{\text{e}}^{-\delta T_b}{w}_k\left(U_b\left(T_b^{-}\right),\left|U_b\left(T_b\right)\right|\right)I_{\left(T_b<\infty ,J=k\right)}|U_b\left(0\right)=u\right]$

其中$\delta \ge 0$ 是贴现因子，$U_b\left(T_b^{-}\right)$ 是破产前盈余，$\left|U_b\left(T_b\right)\right|$ 是破产后赤字，I是示性函数，$w_k$ 是关于破产前盈余和破产后赤字的二元函数，为不失一般性，令$w_k\left(\text{0},\text{0}\right)=\text{1}$ 。同时为确保破产不是必然事件，设定正安全负载条件$E\left[cW-X\right]>0$ 恒成立。故当索赔金额$x>T_i$ 时，下一次索赔的时间服从速率为${\lambda }_1$ 的指数分布，此时若盈余金额$0\le u<b$ ，则其Gerber-Shiu函数为$V_{11}\left(u\right)$ ，若盈余金额$u>b$ ，则其Gerber-Shiu函数为$V_{12}\left(u\right)$ ；当索赔金额$x<T_i$ 时，下一次索赔的时间服从速率为${\lambda }_2$ 的指数分布，此时若盈余金额$0\le u<b$ ，则其Gerber-Shiu函数为$V_{21}\left(u\right)$ ，若盈余金额$u>b$ ，则其Gerber-Shiu函数为$V_{22}\left(u\right)$ 。由于分红是连续性支付的，故盈余过程$U\left(t\right)$ 不会在阈值b处发生跳跃，因此盈余过程$U\left(t\right)$ 在$u=b$ 处连续，而Gerber-Shiu函数是基于盈余过程的期望函数，其连续性由盈余过程的连续性决定，参考文献[7]中Gerber-Shiu函数边界条件的推导方法可得本文边界条件为：${\overrightarrow{V}}_1\left(b^{-}\right)={\overrightarrow{V}}_2\left(b^{+}\right)$ ，${{\overrightarrow{V}}^{\prime }}_1\left(b^{-}\right)={{\overrightarrow{V}}^{\prime }}_2\left(b^{+}\right)$ 。在后续内容中，我们使用以下符号：

${\overrightarrow{V}}_1\left(u\right)=\{\begin{array}{l}{V}_{11}\left(u\right),0\le u<b\\ V_{21}\left(u\right),0\le u<b\end{array}$ ，${\overrightarrow{V}}_2\left(u\right)=\{\begin{array}{l}{V}_{12}\left(u\right),b\le u<\infty \\ V_{22}\left(u\right),b\le u<\infty \end{array}$

3. 积分–微分方程

命题3.1 函数${\overrightarrow{V}}_1\left(u\right)$ ，${\overrightarrow{V}}_2\left(u\right)$ 满足以下的积分–微分方程。

${{\overrightarrow{V}}^{″}}_1\left(u\right)=-\frac{2}{{\sigma }^2}\left[c_1{{\overrightarrow{V}}^{\prime }}_1\left(u\right)+A{\overrightarrow{V}}_1\left(u\right)+{\displaystyle {\int }_0^{u}B}{\overrightarrow{V}}_1\left(u-x\right)\text{d}x+{\overrightarrow{\zeta }}_1\left(u\right)\right]$ (3.1)

${{\overrightarrow{V}}^{″}}_2\left(u\right)=-\frac{2}{{\sigma }^2}\left[c_2{{\overrightarrow{V}}^{\prime }}_2\left(u\right)+A{\overrightarrow{V}}_2\left(u\right)+{\displaystyle {\int }_0^{u-b}B{\overrightarrow{V}}_2\left(u-x\right)\text{d}x}+{\displaystyle {\int }_{u-b}^{u}B{\overrightarrow{V}}_1\left(u-x\right)\text{d}x}+{\overrightarrow{\zeta }}_{\text{1}}\left(u\right)\right]$ (3.2)

其中

$\xi \left(y\right)=\overline{B}\left(y\right)f\left(y\right)$ (3.3)

$\chi \left(y\right)=\overline{H}\left(y\right)b\left(y\right)$ (3.4)

$w_i\left(u\right)={\displaystyle {\int }_u^{\infty }{w}_i\left(u,x-u\right)\xi \left(x\right)\text{d}x}+{\displaystyle {\int }_u^{\infty }{w}_i\left(u,x-u\right)\chi \left(x\right)\text{d}x},u>0$ (3.5)

$A=\left(\begin{array}{cc}-\left({\lambda }_1+\delta \right)& 0\\ 0& -\left({\lambda }_2+\delta \right)\end{array}\right)$ ，$B=\left(\begin{array}{cc}{\lambda }_{\text{1}}\xi \left(x\right)& {\lambda }_{\text{1}}\chi \left(x\right)\\ {\lambda }_{\text{2}}\xi \left(x\right)& {\lambda }_{\text{2}}\chi \left(x\right)\end{array}\right)$ (3.6)

${\overrightarrow{V}}_1\left(u\right)=\left(\begin{array}{c}{V}_{11}\left(u\right)\\ V_{21}\left(u\right)\end{array}\right)$ ，${\overrightarrow{V}}_2\left(u\right)=\left(\begin{array}{c}{V}_{12}\left(u\right)\\ V_{22}\left(u\right)\end{array}\right)$ ，${\overrightarrow{\zeta }}_{\text{1}}\left(u\right)=\left(\begin{array}{c}{\lambda }_1{w}_1\left(u\right)\\ {\lambda }_2{w}_2\left(u\right)\end{array}\right)$ (3.7)

证明 对于$V_{i1}\left(u\right),i=1,2$ ，考虑一个长度为$dt$ 的小时间区间，并以在该区间内可能发生的首次索赔情况为条件，利用全概率公式，有

$\begin{array}{c}{V}_{11}\left(u\right)={\text{e}}^{-\delta dt}\{\left(1-{\lambda }_{1}dt\right)E\left[V_{11}\left(u+c_{1}dt+\sigma W\left(dt\right)\right)\right]\\ +{\lambda }_{1}dt[{\displaystyle {\int }_0^{u+c_{1}dt+\sigma W\left(dt\right)}P\left(X>x\right)}{V}_{11}\left(u+c_{1}dt+\sigma W\left(dt\right)-x\right)f\left(x\right)\text{d}x\\ +{\displaystyle {\int }_0^{u+c_{1}dt+\sigma W\left(dt\right)}P\left(x<T\right)V_{21}\left(u+c_{1}dt+\sigma W\left(dt\right)-x\right)b\left(x\right)\text{d}x}]\\ +{\displaystyle {\int }_{u+c_{1}dt+\sigma W\left(dt\right)}^{\infty }P\left(X>x\right)w_1\left(u+c_{1}dt+\sigma W\left(dt\right),x-u-c_{1}dt-\sigma W\left(dt\right)\right)f}\left(x\right)\text{d}x\\ +{\displaystyle {\int }_{u+c_{1}dt+\sigma W\left(dt\right)}^{\infty }P\left(x<T\right)w_1\left(u+c_{1}dt+\sigma W\left(dt\right),x-u-c_{1}dt-\sigma W\left(dt\right)\right)b}\left(x\right)\text{d}x\}\end{array}$ (3.8)

$\begin{array}{c}{V}_{21}\left(u\right)={\text{e}}^{-\delta dt}\{\left(1-{\lambda }_{2}dt\right)E\left[V_{21}\left(u+c_{1}dt+\sigma W\left(dt\right)\right)\right]\\ +{\lambda }_{2}dt[{\displaystyle {\int }_0^{u+c_{1}dt+\sigma W\left(dt\right)}P\left(X>x\right)}{V}_{11}\left(u+c_{1}dt+\sigma W\left(dt\right)-x\right)f\left(x\right)\text{d}x\\ +{\displaystyle {\int }_0^{u+c_{1}dt+\sigma W\left(dt\right)}P\left(x<T\right)V_{21}\left(u+c_{1}dt+\sigma W\left(dt\right)-x\right)b\left(x\right)\text{d}x}]\\ +{\displaystyle {\int }_{u+c_{1}dt+\sigma W\left(dt\right)}^{\infty }P\left(X>x\right)w_2\left(u+c_{1}dt+\sigma W\left(dt\right),x-u-c_{1}dt-\sigma W\left(dt\right)\right)f}\left(x\right)\text{d}x\\ +{\displaystyle {\int }_{u+c_{1}dt+\sigma W\left(dt\right)}^{\infty }P\left(x<T\right)w_2\left(u+c_{1}dt+\sigma W\left(dt\right),x-u-c_{1}dt-\sigma W\left(dt\right)\right)b}\left(x\right)\text{d}x\}\end{array}$ (3.9)

令$P\left(X>x\right)=\overline{B}\left(x\right)$ ，$P\left(x<T\right)=\overline{H}\left(x\right)$ ，对$E\left[V_{i1}\left(u+c_{1}dt+\sigma w\left(dt\right)\right)\right]$ 应用泰勒展开，并利用布朗运动性质$E\left\{W\left(dt\right)\right\}=0$ 和$E\left\{W^2\left(dt\right)\right\}=dt$ ，可得

$E\left[V_{i1}\left(u+c_{1}dt+\sigma W\left(dt\right)\right)\right]=V_{i1}\left(u\right)+c_1{V^{\prime }}_{i1}\left(u\right)dt+\frac{{\sigma }^2}{2}{V^{″}}_{i1}\left(u\right)dt+o\left(dt\right)$

因此，式(3.8)和式(3.9)可简化为

$\begin{array}{c}{V}_{11}\left(u\right)={\text{e}}^{-\delta dt}\{\left(1-{\lambda }_{1}dt\right)\left(V_{11}\left(u\right)+c_1{V^{\prime }}_{11}\left(u\right)dt+\frac{{\sigma }^2}{2}{V^{″}}_{11}\left(u\right)dt\right)\\ +{\lambda }_{1}dt[{\displaystyle {\int }_0^{u+c_{1}dt+\sigma W\left(dt\right)}\overline{B}\left(x\right)V_{11}\left(u+c_{1}dt+\sigma W\left(dt\right)-x\right)}f\left(x\right)\text{d}x\\ +{\displaystyle {\int }_0^{u+c_{1}dt+\sigma W\left(dt\right)}\overline{H}\left(x\right)V_{21}\left(u+c_{1}dt+\sigma W\left(dt\right)-x\right)b\left(x\right)\text{d}x}]\\ +{\displaystyle {\int }_{u+c_{1}dt+\sigma W\left(dt\right)}^{\infty }\overline{B}\left(x\right)w_1\left(u+c_{1}dt+\sigma W\left(dt\right),x-u-c_{1}dt-\sigma W\left(dt\right)\right)f}\left(x\right)\text{d}x\\ \underset{}{\overset{{}^{}}{}}+{\displaystyle {\int }_{u+c_{1}dt+\sigma W\left(dt\right)}^{\infty }\overline{H}\left(x\right)w_1\left(u+c_{1}dt+\sigma W\left(dt\right),x-u-c_{1}dt-\sigma W\left(dt\right)\right)b}\left(x\right)\text{d}x\}\end{array}$

$\begin{array}{c}{V}_{21}\left(u\right)={\text{e}}^{-\delta dt}\{\left(1-{\lambda }_{2}dt\right)\left(V_{21}\left(u\right)+c_1{V^{\prime }}_{21}\left(u\right)dt+\frac{{\sigma }^2}{2}{V^{″}}_{21}\left(u\right)dt\right)\\ +{\lambda }_{2}dt[{\displaystyle {\int }_0^{u+c_{1}dt+\sigma W\left(dt\right)}\overline{B}\left(x\right)V_{11}\left(u+c_{1}dt+\sigma W\left(dt\right)-x\right)}f\left(x\right)\text{d}x\\ +{\displaystyle {\int }_0^{u+c_{1}dt+\sigma W\left(dt\right)}\overline{H}\left(x\right)V_{21}\left(u+c_{1}dt+\sigma W\left(dt\right)-x\right)b\left(x\right)\text{d}x}]\\ +{\displaystyle {\int }_{u+c_{1}dt+\sigma W\left(dt\right)}^{\infty }\overline{B}\left(x\right)w_1\left(u+c_{1}dt+\sigma W\left(dt\right),x-u-c_{1}dt-\sigma W\left(dt\right)\right)f\left(x\right)\text{d}x}\\ \underset{}{\overset{{}^{}}{}}+{\displaystyle {\int }_{u+c_{1}dt+\sigma W\left(dt\right)}^{\infty }\overline{H}\left(x\right)w_1\left(u+c_{1}dt+\sigma W\left(dt\right),x-u-c_{1}dt-\sigma W\left(dt\right)\right)b}\left(x\right)\text{d}x\}\end{array}$

将等式两边同时除以$dt$ ，并令$dt\to 0$ ，可得

$\begin{array}{c}\left({\lambda }_1+\delta \right)V_{11}\left(u\right)=c_1{V^{\prime }}_{11}\left(u\right)+\frac{{\sigma }^2}{2}{V^{″}}_{11}\left(u\right)+{\lambda }_1[\overline{B}\left(x\right){\displaystyle {\int }_0^u{V}_{11}\left(u-x\right)f\left(x\right)\text{d}x}\\ +{\displaystyle {\int }_0^u\overline{H}\left(x\right)V_{21}\left(u-x\right)b\left(x\right)\text{d}x}]+{\lambda }_1{w}_1\left(u\right)\end{array}$

$\begin{array}{c}\left({\lambda }_2+\delta \right)V_{21}\left(u\right)=c_1{V^{\prime }}_{11}\left(u\right)+\frac{{\sigma }^2}{2}{V^{″}}_{21}\left(u\right)+{\lambda }_2[\overline{B}\left(x\right){\displaystyle {\int }_0^u{V}_{11}\left(u-x\right)f\left(x\right)\text{d}x}\\ +{\displaystyle {\int }_0^u\overline{H}\left(x\right)V_{21}\left(u-x\right)b\left(x\right)\text{d}x}]+{\lambda }_2{w}_2\left(u\right)\end{array}$

最后将式(3.3)~(3.5)代入上式，可以得到

$\begin{array}{c}\left({\lambda }_1+\delta \right)V_{11}\left(u\right)=c_1{V^{\prime }}_{11}\left(u\right)+\frac{{\sigma }^2}{2}{V^{″}}_{11}\left(u\right)+{\lambda }_1[{\displaystyle {\int }_0^u{V}_{11}\left(u-x\right)\xi \left(x\right)\text{d}x}\\ +{\displaystyle {\int }_0^u{V}_{21}\left(u-x\right)\chi \left(x\right)\text{d}x}]+{\lambda }_1{w}_1\left(u\right)\end{array}$

$\begin{array}{c}\left({\lambda }_2+\delta \right)V_{21}\left(u\right)=c_1{V^{\prime }}_{11}\left(u\right)+\frac{{\sigma }^2}{2}{V^{″}}_{21}\left(u\right)+{\lambda }_2[{\displaystyle {\int }_0^u{V}_{11}\left(u-x\right)\xi \left(x\right)\text{d}x}\\ +{\displaystyle {\int }_0^u{V}_{21}\left(u-x\right)\chi \left(x\right)\text{d}x}]+{\lambda }_2{w}_2\left(u\right)\end{array}$

同理，对于$V_{i2}\left(u\right),i=1,2$ ，考虑一个长度为$dt$ 的小时间区间，并以在该区间内可能发生的首次索赔情况为条件，利用全概率公式可以得到

$\begin{array}{c}{V}_{12}\left(u\right)={\text{e}}^{-\delta dt}\{\left(1-{\lambda }_{1}dt\right)E\left[V_{12}\left(u+c_{2}dt+\sigma W\left(dt\right)\right)\right]\\ +{\lambda }_{1}dt[{\displaystyle {\int }_0^{u+c_{2}dt+\sigma W\left(dt\right)-b}P\left(X>x\right)V_{12}(u+c_{2}dt+\sigma W\left(dt\right)-x)f\left(x\right)\text{d}x}\\ +{\displaystyle {\int }_0^{u+c_{2}dt+\sigma W\left(dt\right)-b}P\left(x<T\right)V_{22}(u+c_{2}dt+\sigma W\left(dt\right)-x)b\left(x\right)\text{d}x}\\ +{\displaystyle {\int }_{u+c_{2}dt+\sigma W\left(dt\right)-b}^{u+c_{2}dt+\sigma W\left(dt\right)}P\left(X>x\right)V_{11}\left(u+c_{2}dt+\sigma W\left(dt\right)-x\right)f\left(x\right)\text{d}x}\\ +{\displaystyle {\int }_{u+c_{2}dt+\sigma W\left(dt\right)-b}^{u+c_{2}dt+\sigma W\left(dt\right)}P\left(x<T\right)V_{21}\left(u+c_{2}dt+\sigma W\left(dt\right)-x\right)b\left(x\right)\text{d}x}]\\ +{\displaystyle {\int }_{u+c_{1}dt+\sigma W\left(dt\right)}^{\infty }P\left(X>x\right)w_1\left(u+c_{1}dt+\sigma W\left(dt\right),x-u-c_{1}dt-\sigma W\left(dt\right)\right)f\left(x\right)\text{d}x}\\ +{\displaystyle {\int }_{u+c_{1}dt+\sigma W\left(dt\right)}^{\infty }P\left(x<T\right)w_1\left(u+c_{1}dt+\sigma W\left(dt\right),x-u-c_{1}dt-\sigma W\left(dt\right)\right)b\left(x\right)\text{d}x}\}\end{array}$ (3.10)

$\begin{array}{c}{V}_{22}\left(u\right)={\text{e}}^{-\delta dt}\{\left(1-{\lambda }_{2}dt\right)E\left[V_{22}\left(u+c_{2}dt+\sigma W\left(dt\right)\right)\right]\\ +{\lambda }_{2}dt[{\displaystyle {\int }_0^{u+c_{2}dt+\sigma W\left(dt\right)-b}P\left(X>x\right)V_{12}(u+c_{2}dt+\sigma W\left(dt\right)-x)f\left(x\right)\text{d}x}\\ +{\displaystyle {\int }_0^{u+c_{2}dt+\sigma W\left(dt\right)-b}P\left(x<T\right)V_{22}(u+c_{2}dt+\sigma W\left(dt\right)-x)b\left(x\right)\text{d}x}\\ +{\displaystyle {\int }_{u+c_{2}dt+\sigma W\left(dt\right)-b}^{u+c_{2}dt+\sigma W\left(dt\right)}P\left(X>x\right)V_{11}\left(u+c_{2}dt+\sigma W\left(dt\right)-x\right)f\left(x\right)\text{d}x}\\ +{\displaystyle {\int }_{u+c_{2}dt+\sigma W\left(dt\right)-b}^{u+c_{2}dt+\sigma W\left(dt\right)}P\left(x<T\right)V_{21}\left(u+c_{2}dt+\sigma W\left(dt\right)-x\right)b\left(x\right)\text{d}x}]\\ +{\displaystyle {\int }_{u+c_{1}dt+\sigma W\left(dt\right)}^{\infty }P\left(X>x\right)w_2\left(u+c_{1}dt+\sigma W\left(dt\right),x-u-c_{1}dt-\sigma W\left(dt\right)\right)f\left(x\right)\text{d}x}\\ +{\displaystyle {\int }_{u+c_{1}dt+\sigma W\left(dt\right)}^{\infty }P\left(x<T\right)w_2\left(u+c_{1}dt+\sigma W\left(dt\right),x-u-c_{1}dt-\sigma W\left(dt\right)\right)b\left(x\right)\text{d}x}\}\end{array}$ (3.11)

令$P\left(X>x\right)=H\left(x\right)$ ，$P\left(x<T\right)=1-H\left(x\right)=\overline{H}\left(x\right)$ ，对$E\left[V_{i2}\left(u+c_{1}dt+\sigma w\left(dt\right)\right)\right]$ 应用泰勒展开，并利用布朗运动性质$E\left\{W\left(dt\right)\right\}=0$ 和$E\left\{W^2\left(dt\right)\right\}=dt$ ，可以得到

$E\left[V_{i2}\left(u+c_{1}dt+\sigma W\left(dt\right)\right)\right]=V_{i2}\left(u\right)+c_1{V^{\prime }}_{i2}\left(u\right)dt+\frac{{\sigma }^2}{2}{V^{″}}_{i2}\left(u\right)dt+o\left(dt\right)$

因此，式(3.10)和式(3.11)可简化为

$\begin{array}{c}{V}_{12}\left(u\right)={\text{e}}^{-\delta dt}\{\left(1-{\lambda }_{1}dt\right)\left(V_{12}\left(u\right)+c_2{V^{\prime }}_{12}\left(u\right)dt+\frac{{\sigma }^2}{2}{V^{″}}_{12}\left(u\right)dt\right)\\ +{\lambda }_{1}dt[{\displaystyle {\int }_0^{u+c_{2}dt+\sigma W\left(dt\right)-b}\overline{B}\left(x\right)V_{12}\left(u+c_{2}dt+\sigma W\left(dt\right)-x\right)f\left(x\right)\text{d}x}\\ +{\displaystyle {\int }_0^{u+c_{1}dt+\sigma W\left(dt\right)}\overline{H}\left(x\right)V_{21}\left(u+c_{2}dt+\sigma W\left(dt\right)-x\right)b\left(x\right)\text{d}x}\\ +{\displaystyle {\int }_{u+c_{2}dt+\sigma W\left(dt\right)-b}^{u+c_{2}dt+\sigma W\left(dt\right)}\overline{B}\left(x\right)V_{11}\left(u+c_{2}dt+\sigma W\left(dt\right)-x\right)f\left(x\right)\text{d}x}\\ +{\displaystyle {\int }_{u+c_{2}dt+\sigma W\left(dt\right)-b}^{u+c_{2}dt+\sigma W\left(dt\right)}\overline{H}\left(x\right)V_{11}\left(u+c_{2}dt+\sigma W\left(dt\right)-x\right)b\left(x\right)\text{d}x}\\ +{\displaystyle {\int }_{u+c_{1}dt+\sigma W\left(dt\right)}^{\infty }\overline{B}\left(x\right)w_1\left(u+c_{1}dt+\sigma W\left(dt\right),x-u-c_{1}dt-\sigma W\left(dt\right)\right)f\left(x\right)\text{d}x}\\ +{\displaystyle {\int }_{u+c_{1}dt+\sigma W\left(dt\right)}^{\infty }\overline{H}\left(x\right)w_1\left(u+c_{1}dt+\sigma W\left(dt\right),x-u-c_{1}dt-\sigma W\left(dt\right)\right)b\left(x\right)\text{d}x}\}\end{array}$

$\begin{array}{c}{V}_{22}\left(u\right)={\text{e}}^{-\delta dt}\{\left(1-{\lambda }_{2}dt\right)\left(V_{22}\left(u\right)+c_2{V}^{\text{'}}{}_{22}\left(u\right)dt+\frac{{\sigma }^2}{2}{V}^{\text{'}\text{'}}{}_{22}\left(u\right)dt\right)\\ +{\lambda }_{2}dt[{\displaystyle {\int }_0^{u+c_{2}dt+\sigma W\left(dt\right)-b}\overline{B}\left(x\right)V_{12}\left(u+c_{2}dt+\sigma W\left(dt\right)-x\right)f\left(x\right)\text{d}x}\\ +{\displaystyle {\int }_0^{u+c_{1}dt+\sigma W\left(dt\right)}\overline{H}\left(x\right)V_{21}\left(u+c_{2}dt+\sigma W\left(dt\right)-x\right)b\left(x\right)\text{d}x}\\ +{\displaystyle {\int }_{u+c_{2}dt+\sigma W\left(dt\right)-b}^{u+c_{2}dt+\sigma W\left(dt\right)}\overline{B}\left(x\right)V_{11}\left(u+c_{2}dt+\sigma W\left(dt\right)-x\right)f\left(x\right)\text{d}x}\\ +{\displaystyle {\int }_{u+c_{2}dt+\sigma W\left(dt\right)-b}^{u+c_{2}dt+\sigma W\left(dt\right)}\overline{H}\left(x\right)V_{11}\left(u+c_{2}dt+\sigma W\left(dt\right)-x\right)b\left(x\right)\text{d}x}\\ +{\displaystyle {\int }_{u+c_{1}dt+\sigma W\left(dt\right)}^{\infty }\overline{B}\left(x\right)w_1\left(u+c_{1}dt+\sigma W\left(dt\right),x-u-c_{1}dt-\sigma W\left(dt\right)\right)f\left(x\right)\text{d}x}\\ +{\displaystyle {\int }_{u+c_{1}dt+\sigma W\left(dt\right)}^{\infty }\overline{H}\left(x\right)w_1\left(u+c_{1}dt+\sigma W\left(dt\right),x-u-c_{1}dt-\sigma W\left(dt\right)\right)b\left(x\right)\text{d}x}\}\end{array}$

然后将等式两边除以$dt$ ，并令$dt\to 0$ ，可得

$\begin{array}{c}\left({\lambda }_1+\delta \right)V_{12}\left(u\right)=c_2{V^{\prime }}_{12}\left(u\right)+\frac{{\sigma }^2}{2}{V^{″}}_{12}\left(u\right)+{\lambda }_1[{\displaystyle {\int }_0^{u-b}\overline{B}\left(x\right)V_{12}\left(u-x\right)f\left(x\right)\text{d}x}\\ +{\displaystyle {\int }_0^{u-b}\overline{H}\left(x\right)V_{22}\left(u-x\right)b\left(x\right)\text{d}x}+{\displaystyle {\int }_{u-b}^u\overline{B}\left(x\right)V_{11}\left(u-x\right)f\left(x\right)\text{d}x}\\ +{\displaystyle {\int }_{u-b}^u\overline{H}\left(x\right)V_{21}\left(u-x\right)b\left(x\right)\text{d}x}]+{\lambda }_1{w}_1\left(u\right)\end{array}$

$\begin{array}{c}\left({\lambda }_2+\delta \right)V_{22}\left(u\right)=c_2{V^{\prime }}_{22}\left(u\right)+\frac{{\sigma }^2}{2}{V^{″}}_{22}\left(u\right)+{\lambda }_1[{\displaystyle {\int }_0^{u-b}\overline{B}\left(x\right)V_{12}\left(u-x\right)f\left(x\right)\text{d}x}\\ +{\displaystyle {\int }_0^{u-b}\overline{H}\left(x\right)V_{22}\left(u-x\right)b\left(x\right)\text{d}x}+{\displaystyle {\int }_{u-b}^u\overline{B}\left(x\right)V_{11}\left(u-x\right)f\left(x\right)\text{d}x}\\ +{\displaystyle {\int }_{u-b}^u\overline{H}\left(x\right)V_{21}\left(u-x\right)b\left(x\right)\text{d}x}]+{\lambda }_2{w}_2\left(u\right)\end{array}$

最后将${V^{″}}_{i1}\left(u\right),{V^{″}}_{i2}\left(u\right)\left(i=1,2\right)$ 移到等号左边，整理可得

$\begin{array}{c}{V^{″}}_{11}\left(u\right)=\left(-\frac{2}{{\sigma }^2}\right)[c_1{V^{\prime }}_{11}\left(u\right)-\left({\lambda }_1+\delta \right)V_{11}\left(u\right)+{\lambda }_1{\displaystyle {\int }_0^u{V}_{11}\left(u\right)\xi \left(x\right)\text{d}x}\\ +{\lambda }_1{\displaystyle {\int }_0^u{V}_{21}\left(u-x\right)\chi \left(x\right)\text{d}x}]+{\lambda }_1{w}_1\left(u\right)\end{array}$

$\begin{array}{c}{V^{″}}_{21}\left(u\right)=\left(-\frac{2}{{\sigma }^2}\right)[c_2{V^{\prime }}_{21}\left(u\right)-\left({\lambda }_2+\delta \right)V_{21}\left(u\right)+{\lambda }_2{\displaystyle {\int }_0^u{V}_{11}\left(u\right)\xi \left(x\right)\text{d}x}\\ +{\lambda }_2{\displaystyle {\int }_0^u{V}_{21}\left(u-x\right)\chi \left(x\right)\text{d}x}]+{\lambda }_2{w}_2\left(u\right)\end{array}$

$\begin{array}{c}{V^{″}}_{12}\left(u\right)=\left(-\frac{2}{{\sigma }^2}\right)[c_2{V^{\prime }}_{12}\left(u\right)-\left({\lambda }_1+\delta \right)V_{12}\left(u\right)\\ +{\lambda }_1[{\displaystyle {\int }_0^{u-b}{V}_{12}\left(u-x\right)\xi \left(x\right)\text{d}x}+{\displaystyle {\int }_0^{u-b}{V}_{22}\left(u-x\right)\chi \left(x\right)\text{d}x}\\ +{\displaystyle {\int }_{u-b}^u{V}_{11}\left(u-x\right)\xi \left(x\right)\text{d}x}+{\displaystyle {\int }_{u-b}^u{V}_{21}\left(u-x\right)\chi \left(x\right)\text{d}x}]+{\lambda }_1{w}_1\left(u\right)\end{array}$

$\begin{array}{c}{V^{″}}_{22}\left(u\right)=\left(-\frac{2}{{\sigma }^2}\right)[c_2{V^{\prime }}_{22}\left(u\right)-\left({\lambda }_2+\delta \right)V_{22}\left(u\right)\\ +{\lambda }_2[{\displaystyle {\int }_0^{u-b}{V}_{12}\left(u-x\right)\xi \left(x\right)\text{d}x}+{\displaystyle {\int }_0^{u-b}{V}_{22}\left(u-x\right)\chi \left(x\right)\text{d}x}\\ +{\displaystyle {\int }_{u-b}^u{V}_{11}\left(u-x\right)\xi \left(x\right)\text{d}x}+{\displaystyle {\int }_{u-b}^u{V}_{21}\left(u-x\right)\chi \left(x\right)\text{d}x}]+{\lambda }_2{w}_2\left(u\right)\end{array}$

将式(2.6)和式(2.7)代入以上四个公式，即可得式(2.1)和式(2.2)。

4. 数值分析

传统独立模型假设索赔金额大小与下一次索赔间隔完全独立，即索赔间隔服从$\text{Exp}\left(\lambda \right)$ ，其中$\lambda =\frac{{\lambda }_1+{\lambda }_2}{2}$ ，设其破产概率为$\psi \left(u\right)$ ，在本节中我们考虑不同参数的数值示例来对比独立模型与相依模型。假设分红阈值$b=10$ ，分红速率$d=0.4$ ，保费率$c=2.2$ ，布朗运动扩散率$\sigma =0.7$ ，索赔额$X_i$ 服从指数分布$X~\text{Exp}\left(0.25\right)$ ，索赔阈值$T_i=4$ ，实际赔付金额为$Z=\min \left(X,T\right)$ ，${\lambda }_2=0.35$ 。

当${\lambda }_1>{\lambda }_2$ 时，固定低赔付场景索赔频率${\lambda }_2=0.35$ ，逐步增大高赔付场景索赔频率${\lambda }_1$ ，${\lambda }_1$ 分别取值为0.35，0.5，0.65，0.8，0.95，其中${\lambda }_1=0.35$ 对应独立模型$\lambda ={\lambda }_1={\lambda }_2$ ，设初始盈余$u=5$ ，则不同${\lambda }_1$ 下相依模型与独立模型的破产概率对比为下表1。

Table 1. Table of ruin probabilities for positive dependent model and independent model

表1. 正相依模型与独立模型的破产概率表

${\lambda }_1$	${\lambda }_2$	${\lambda }_1-{\lambda }_2$	$V\left(5\right)$	$\psi \left(5\right)$	风险偏差
0.35	0.35	0	0.178	0.178	0
0.50	0.35	0.15	0.234	0.178	0.056
0.65	0.35	0.30	0.291	0.178	0.113
0.80	0.35	0.45	0.348	0.178	0.170
0.95	0.35	0.60	0.405	0.178	0.227

当${\lambda }_1<{\lambda }_2$ 时，固定低赔付场景索赔频率${\lambda }_2=0.35$ ，逐步减小高赔付场景索赔频率${\lambda }_1$ ，${\lambda }_1$ 分别取值为0.35，0.25，0.15，0.05，0，其中${\lambda }_1=0.35$ 对应独立模型$\lambda ={\lambda }_1={\lambda }_2$ ，设初始盈余$u=5$ ，则不同${\lambda }_1$ 下相依模型与独立模型的破产概率对比为下表2。

Table 2. Table of ruin probabilities for negative dependent models and independent models

表2. 负相依模型与独立模型的破产概率表

${\lambda }_1$	${\lambda }_2$	${\lambda }_1-{\lambda }_2$	$V\left(5\right)$	$\psi \left(5\right)$	风险偏差
0.35	0.35	0	0.178	0.178	0
0.25	0.35	0.10	0.146	0.178	−0.032
0.15	0.35	0.20	0.117	0.178	−0.061
0.05	0.35	0.30	0.092	0.178	−0.086
0	0.35	0.35	0.071	0.178	−0.107

由以上两表可知，与相依模型相比，独立模型对破产概率存在一定的误差，与正向相依模型对比，独立模型会低估破产风险，与反向相依模型对比，独立模型会高估破产风险，故相依模型更具符合实际。