数学物理学报, 2023, 43(3): 957-969

基于Ornstein-Uhlenbeck过程下具有两个再保险公司的比例再保险与投资

黄玲,1, 刘海燕1,2, 陈密,1,2,*

1福建师范大学数学与统计学院 福州350117

2福建省分析数学及应用重点实验室 福州350117

Proportional Reinsurance and Investment Based on the Ornstein-Uhlenbeck Process in the Presence of Two Reinsurers

Huang ,1, Liu Haiyan1,2, Chen Mi,1,2,*

1School of Mathematics and Statistics, Fujian Normal University, Fuzhou 350117

2Fujian Provincial Key Laboratory of Mathematical Analysis and its Applications, Fuzhou 350117

通讯作者: *陈密,E-mail: chenmi0610@163.com

收稿日期: 2022-04-25   修回日期: 2023-02-6  

基金资助: 国家自然科学基金(11701087)
福建省自然科学基金(2019J01673)

Received: 2022-04-25   Revised: 2023-02-6  

Fund supported: NSFC(11701087)
NSF of Fujian Province (2019J01673)

作者简介 About authors

黄玲,E-mail:hl2193088930@163.com

摘要

该文研究了两类风险模型下具有两个再保险公司的最优再保险和投资问题.保险公司购买比例再保险并投资于无风险资产和风险资产组成的金融市场,其风险资产价格模型受Ornstein-Uhlenbeck过程影响.假设再保险的保费按照指数保费原则来计算,保险公司的目标是使终端财富的期望指数效用最大化.利用随机控制理论和HJB方程,推导出了最优策略和值函数的显式表达式.最后,通过数值分析验证了模型参数对最优策略的影响.

关键词： Ornstein-Uhlenbeck过程; 指数效用; 比例再保险; 投资; 指数保费原则

Abstract

This paper studies the optimal reinsurance and investment problem with two reinsurance companies under two risk models. The insurance company purchases proportional reinsurance and invests in the financial market consisting of one risk-free asset and one risky asset, where the price of the risky asset is influenced by the Ornstein-Uhlenbeck process. Assuming that premiums for reinsurance are calculated according to the exponential premium principle, and the insurer's goal is to maximize the expected exponential utility of terminal wealth. Using stochastic control theory and HJB equation, the explicit expressions of the optimal strategy and value function are derived. Finally, the influence of model parameters on optimal strategy is verified by numerical analysis.

Keywords： Ornstein-Uhlenbeck process; Exponential utility; Proportional reinsurance; Investment; Exponential premium principle

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本文引用格式

黄玲, 刘海燕, 陈密. 基于Ornstein-Uhlenbeck过程下具有两个再保险公司的比例再保险与投资[J]. 数学物理学报, 2023, 43(3): 957-969

Huang , Liu Haiyan, Chen Mi. Proportional Reinsurance and Investment Based on the Ornstein-Uhlenbeck Process in the Presence of Two Reinsurers[J]. Acta Mathematica Scientia, 2023, 43(3): 957-969

1 引言

投资和再保险是保险风险理论中的两类重要问题,长期以来一直受到许多学者的关注.具体来说,人们通常考虑通过购买再保险来控制保险公司所面临的风险,并将盈余财富投资于金融市场以使某种价值目标最大化. 随机控制理论和Hamilton-Jacobi-Bellman(HJB)方程为解决此类问题提供了强有力的工具.相关文献可见文献[1⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓-12].

为了数学上的方便,通常假设保费是通过期望值准则计算的.然而,对于均值相同的两个风险仍然可能存在较大的差异, 那么收取的相关保费也应该不同.指数保费原则,即所谓的零效用原则,在保险数学和精算实践中发挥着重要作用.它有很多很好的特性,包括独立风险的可加性.它在数理金融中也被广泛应用于市场上各种保险产品的定价,见参考文献[13⇓⇓⇓-17]. 对于其他保费准则下的比例再保险问题,见文献[18⇓-20].

在实践中,保险公司经常购买再保险,以降低其保险组合的风险.为简单起见,文献中通常假设一个保险公司只能从一个再保险公司购买再保险. 然而,常见的情况是,保险公司希望通过向不同风险厌恶的多家再保险公司购买再保险来分散风险.因此,研究具有多个再保险公司的最优再保险模型具有重要意义.例如, Chi和Meng[21]研究了在保险人总风险暴露的风险值或条件风险值最小化准则下,具有多个再保险人的最优再保险问题; Chen和Yuen[22]研究了扩散逼近模型下受交易费用影响的最优分红和有两个再保险公司的比例再保险问题; Meng等[23]研究了在连续时间模型下的最优再保险问题; Yao和Fan[24] 假设保险公司能够动态地控制分红、再融资和再保险策略,利用最优控制方法,研究了保险公司值函数最大化的最优策略,以及交易成本和最终价值对破产的影响.

在大多数文献中,在研究投资问题时都假设风险资产的价格遵循几何布朗运动.为了更好地刻画风险资产的某些特征,比几何布朗运动更一般的风险资产价格模型近年来受到越来越多的关注.受Ornstein-Uhlenbeck过程影响下的风险资产价格模型,可参考文献[25⇓⇓-28],受其他模型影响的参考文献[29⇓-31].

在本文中,在风险资产价格模型受Ornstein-Uhlenbeck过程影响下,研究了保险公司最优投资和比例再保险策略. 与Liang等[26]不同的是,本文采用两个再保险公司来研究再保险问题,对于再保险保费是依据不同参数的指数保费原则计算的,并且分析了最优策略随模型参数的变化规律.比较有意思的是, Liang等[26]在复合泊松风险模型下的结果比扩散逼近风险模型下的结果复杂,而本文在扩散逼近风险模型下的结果比复合泊松风险模型下的结果复杂.论文的其余部分组织如下:第2节给出了模型和假设.第3节在复合泊松风险模型下推导出了最优投资和再保险策略的显式表达式.第4节在扩散逼近风险模型下推导出了最优投资和再保险策略的显式表达式. 第5节对最优投资和再保险策略进行了数值分析.

2 模型介绍

在经典风险模型下,盈余过程$X_t$由下式给出

$\begin{equation}\label{eq:a1} X_t=u+ct-S_t=u+ct-\sum\limits_{i=1}^{N_t}Y_i, \end{equation} $

其中$u$为初始盈余, $c$为单位时间的保险费率,索赔数过程$N_t$是一个强度为$\lambda$的齐次泊松过程, $\{Y_i,i\geq1\}$是一个正的、独立同分布的索赔额随机变量序列,具有共同的分布$F(y)$,用$\mu=E(Y_i)$和$M_Y(r)=E(e^{rY_i})$分别表示$Y$的均值和矩母函数.通常,我们假设索赔数过程$N_t$独立于索赔额$Y_i$,并且存在$0<\zeta\leq+\infty$使得$\lim_{r\rightarrow\zeta}E(Ye^{rY})=+\infty$,存在$0<r<\zeta$使得$E(Ye^{rY})=M_Y'(r)$.

考虑具有两个再保险公司的比例再保险问题.对于给定的自留比例$q\in(0,1)$,将总索赔$S_t$按比例分成三部分:保险公司占比$q$;第一个再保险公司占比$u(1-q)$;第二个再保险公司占比$(1-u)(1-q)$,其中$q,u\in(0,1)$.那么,保险公司应付索赔额为$S_I(t)=\sum\limits_{i=1}^{N_t}qY_i$,第一个再保险公司应付索赔额为$S_1(t)=\sum\limits_{i=1}^{N_t}u(1-q)Y_i$,第二个再保险公司应付索赔额为$S_2(t)=\sum\limits_{i=1}^{N_t}(1-u)(1-q)Y_i$.此外,保险公司被允许将其盈余投资于由无风险资产(债券)和风险资产(股票)组成的金融市场.具体来说,无风险资产的价格过程由下式给出

${\rm d}R_t=rR_t{\rm d}t,\quad r>0,$

其中$r$是无风险利率.一个常用的股票价格模型是它遵循几何布朗运动,即股票的价格$P_t$满足一个随机微分方程

${\rm d}P_t=aP_t{\rm d}t+\sigma P_t{\rm d}W_t^{(1)},$

其中, $a(>r)$和$\sigma$是正常数,分别代表风险资产的预期瞬时收益率和风险资产价格的波动性, $W_t^{(1)}$是标准布朗运动.参照Liang等[26],本文考虑了一个具有牛市和熊市特征的股票价格模型,即

$\begin{equation}\label{eq:a2} {\rm d}P_t=P_t(a(t){\rm d}t+\sigma {\rm d}W_t^{(1)}), \end{equation}$

其中, $a(t)=a+m(t)$, $m(t)$是$dm(t)=\alpha m(t){\rm d}t+\beta {\rm d}W_t^{(2)}$的解, $m(0)=m_0$是一个任意常数, $W_t^{(2)}$是另一个标准布朗运动.用$\rho_1$表示$W_t^{(1)}$和$W_t^{(2)}$的相关系数,即$E(W_t^{(1)}W_t^{(2)})=\rho_1t$.其中$a$, $\sigma$, $\alpha$和$\beta$是已知的常数,且$a$和$\sigma$都是正的.对于这个模型,如果股票有一个目标平均增长率,并且有一段时间$m(t)$显著大于0,那么这可以被认为是牛市.相反,当$m(t)$明显小于0时,这可以被认为是熊市.

设$A_t$为$t$时刻投资于风险资产的资金总额.遵循Browne[1] 的假设,我们允许$A_t<0$和$A_t>X_t$.换句话说,我们允许公司卖空风险资产$(A_t<0)$,也允许公司借钱投资风险资产$(A_t>X_t)$.假设保险公司可以连续地购买再保险, $t$时刻支付给两个再保险公司的保费率分别为$\delta_1(q_t,u_t)$和$\delta_2(q_t,u_t)$.未投资于风险资产的所有资金都被投资于无风险资产,因此带投资和再保险策略的盈余过程为

$\begin{eqnarray*} {\rm d}X_t&=&A_t\frac{{\rm d}P_t}{P_t}+(X_t-A_t )\frac{{\rm d}R_t}{R_t}+(c-\delta_1(q_t,u_t)-\delta_2(q_t,u_t)){\rm d}t-q_t{\rm d}S_t \nonumber\\ & =&[rX_t+(m(t)+a-r)A_t+(c-\delta_1(q_t,u_t)-\delta_2(q_t,u_t))]{\rm d}t+A_t \sigma {\rm d}W_t^{(1)}-q_t{\rm d}S_t. \label{eq:a3} \end{eqnarray*}$

在本文中,我们假设再保险的保费根据指数保费原则进行计算.具体地,对于风险$U$,所收取的保费$\pi_b(U)$为

$\pi_b(U)=\frac{1}{b}\log E(e^{bU}),$

其中常数$b>0$用于衡量再保险公司的风险厌恶,且$\pi_b(U)$关于$b$严格单增.那么

$\delta_1(q,u)=\beta_1^{-1}\log E(e^{\beta_1 S_1(t)})/t=\beta_1^{-1}\lambda[M_Y(\beta_1 u(1-q))-1],$
$\delta_2(q,u)=\beta_2^{-1}\log E(e^{\beta_2 S_2(t)})/t=\beta_2^{-1}\lambda[M_Y(\beta_2 (1-u)(1-q))-1],$

其中,$\beta_1\mbox{、}\beta_2>0$分别是两个再保险公司的安全载荷.则盈余过程(2.3)式可以改写为

$\begin{equation}\label{eq:a4} \left\{ \begin{array}{lll} {\rm d}X_t=[rX_t+(m(t)+a-r)A_t+c-\lambda(\beta_1^{-1}(M_Y(\beta_1 u_t(1-q_t))-1)+\beta_2^{-1}\\ \qquad \times(M_Y(\beta_2 (1-u_t)(1-q_t))-1))]{\rm d}t+A_t \sigma {\rm d}W_t^{(1)}-q_t{\rm d}S_t,\ (A.,q.,u.)\in \Pi, \\ dm(t)= \alpha m(t){\rm d}t+\beta {\rm d}W_t^{(2)}, m(0)=m_0. \end{array} \right. \end{equation}$

${\bf注2.1} $ 本文假设连续交易是允许的,并且所有资产都是无限可分的.此外,我们在完全概率空间 $(\Omega,{\cal F},P)$上运算, 在该空间上过程$X_t$被很好地定义.时间$t$的信息由$X_t$生成的完整滤流${\cal F}_t$给出.可容许策略 $(A_t,q_t,u_t)$是${\cal F}_t$可测的,并且满足$E[\int_{0}^{T} A_s^{2}{\rm d}s]<+\infty, \forall T<\infty$. 所有可容许策略的集合用$\Pi$表示.

这里,我们假设保险公司有一个指数效用函数,其目标是最大化终端时刻$T$的财富效用.因此,我们的研究问题可以归结为求解值函数

$\begin{equation}\label{eq:a5} V(t,x,m)=\max\limits_{(A,q,u)\in\Pi} E[u(X_T)\mid(X_t,m(t))=(x,m)], \end{equation}$

其中, $u(x)=\lambda_1-\frac{\eta}{\nu}e^{-\nu x}$, $\eta>0$, $\nu>0$.该效用具有恒定的绝对风险厌恶参数$\nu$.

设$C^{(1,2,2)}$是$\phi(t,x,m)$的空间,使得$\phi$及其偏导数$\phi_t$, $\phi_x$, $\phi_m$, $\phi_{xx}$, $\phi_{xm}$和$\phi_{mm}$在$[T]\times R\times R$上是连续的.根据经典的随机控制理论方法易知, 如果值函数$V\in C^{(1,2,2)}$,那么$V$满足下面的HJB方程

$\begin{equation}\label{eq:a6} \sup\limits_{(A,q,u)\in\Pi}{\cal A}^{A,q,u} V(t,x,m)=0, t<T, \end{equation}$

其终端条件为

$\begin{equation}\label{eq:a7} V(T,x,m)=u(x), \end{equation}$

其中

$\begin{eqnarray*} {\cal A}^{A,q,u} V(t,x,m)&=&V_t+[rx+(m+a-r)A+c-\lambda(\beta_1^{-1}(M_Y(\beta_1 u(1-q))-1) \nonumber\\ && +\beta_2^{-1}(M_Y(\beta_2 (1-u)(1-q))-1))]V_x+\alpha mV_m+\frac{1}{2}A^2\sigma^2 V_{xx}+\frac{1}{2} \beta^2 V_{mm} \nonumber\\ && + A\sigma \beta \rho_1 V_{xm}+\lambda E[V(t,x-qY,m)-V(t,x,m)]. \end{eqnarray*}$

利用Yang和Zhang[32]的标准方法,可以得到如下验证定理.

${\bf定理2.1}$ 设$W\in C^{(1,2,2)}$是满足(2.6)式和(2.7)式的经典解.则由(2.5)式给出的值函数$V$与$W$相等. 此外,若存在$(A^{*},q^{*},u^{*})\in\Pi$,使得

${\cal A}^{A^{*},q^{*},u^{*}} V(t,x,m)=0,$

对$\forall(t,x,m)\in[0,T)\times R\times R$成立.则策略$(A^{*}(t,X_t^{*},m(t)),q^{*} (t,X_t^{*},m(t)),u^{*} (t,X_t^{*},m(t)))$为最优的投资和再保险策略,即$E[u(X_T^{*})\mid(X_t^{*},m(t))=(x,m)]=V(t,x,m),$ 这里$X_t^{*}$是最优策略下的盈余过程.

3 复合泊松风险模型下的最优策略和值函数

借鉴Browne[1] 或Yang和Zhang[32]的方法,将$V(t,x,m)$拆成一个很好的形式

$\begin{equation}\label{eq:b1} V(t,x,m)=\lambda_1-\frac{\eta}{\nu} e^{-\nu xe^{r(T-t)} +G(t,m)} \end{equation}$

来求解方程(2.6),其中$G(t,m)$是一个合适的函数,使得(3.1)式是方程(2.6)的解.同样,终端条件$V(T,x,m)=u(x)$意味着

$\begin{equation}\label{eq:b2} G(T,m)=0. \end{equation}$

设$G_t$, $G_m$, $G_{mm}$是$G(t,m)$的偏导数.根据(3.1),我们得到

$\begin{matrix}\label{eq:b3} \left\{ \begin{array}{lll} V_t=[V(t,x,m)-\lambda_1 ][\nu xre^{r(T-t)}+G_t],\\ V_x=[V(t,x,m)-\lambda_1 ][-\nu e^{r(T-t)}],\\ V_m=[V(t,x,m)-\lambda_1 ][G_m],\\ V_{xx}=[V(t,x,m)-\lambda_1 ][\nu^{2} e^{2r(T-t)}],\\ V_{mm}=[V(t,x,m)-\lambda_1 ][G_m^{2}+G_{mm}],\\ V_{xm}=[V(t,x,m)-\lambda_1 ][-\nu e^{r(T-t)} G_m],\\ E[V(t,x-qY,m)-V(t,x,m)]=[V(t,x,m)-\lambda_1 ][M_Y(\nu qe^{r(T-t)})-1].\\ \end{array} \right. \end{matrix}$

将(3.3)式带入方程(2.6),且由$V(t,x,m)-\lambda_1<0$可得

$\begin{eqnarray*}\label{eq:b4} &&G_t-c\nu e^{r(T-t)}+\alpha mG_m+\frac{1}{2} \beta^2 (G_m^{2}+G_{mm})-\lambda[(\beta_1^{-1}+\beta_2^{-1})\nu e^{r(T-t)}+1] \nonumber\\ && +\inf\limits_{A}\{[-(m+a-r)A+\frac{1}{2}A^2\sigma^2\nu e^{r(T-t)}-A\sigma \beta \rho_1G_m]\nu e^{r(T-t)}\} \nonumber\\ && +\inf\limits_{q,u}\{ \lambda[\beta_1^{-1}M_Y(\beta_1u(1-q))+\beta_2^{-1}M_Y(\beta_2(1-u)(1-q))] \nu e^{r(T-t)} \nonumber\\ && +\lambda M_Y(\nu qe^{r(T-t)})\}=0. \end{eqnarray*}$

设

$f_1(A,t)=[-(m+a-r)A+\frac{1}{2}A^2\sigma^2\nu e^{r(T-t)}-A\sigma \beta \rho_1G_m]\nu e^{r(T-t)},$
$f_2(q,u,t)=\lambda[\beta_1^{-1}M_Y(\beta_1u(1-q))+\beta_2^{-1}M_Y(\beta_2(1-u)(1-q))] \nu e^{r(T-t)}+\lambda M_Y(\nu qe^{r(T-t)}).$

对$f_1(A,t)$中的$A$求一阶偏导,令$\frac{\partial f_1(A,t)}{\partial A}=0$,得到

$\begin{eqnarray*} A_t^*=\frac{(m+a-r)+\beta\sigma\rho_1G_m}{\sigma^2\nu e^{r(T-t)}}. \end{eqnarray*}$

分别对$f_2(q,u,t)$中的$q\mbox{、}\,u$求偏导,易知 $\frac{\partial^2 f_2(q,u,t)}{\partial q^2}>0\mbox{、}\, \frac{\partial^2 f_2(q,u,t)}{\partial u^2}>0$.令$\frac{\partial f_2(q,u,t)}{\partial q}=0$, $\frac{\partial f_2(q,u,t)}{\partial u}=0$,得

$M_Y'(\nu qe^{r(T-t)})=M_Y'(\beta_1u(1-q))=M_Y'(\beta_2(1-u)(1-q)).$

进而可得$f_2(q,u,t)$的最小值点为

$\begin{eqnarray*} \left\{ \begin{array}{lll} u_t^*=\frac{\beta_2}{\beta_1+\beta_2},\\[3mm] q_t^*=\frac{\beta_1\beta_2}{(\beta_1+\beta_2)\nu e^{r(T-t)}+\beta_1\beta_2}. \end{array} \right. \end{eqnarray*}$

将$(A_t^*,q_t^*,u_t^*)$代入方程(3.4)中得

$\begin{eqnarray*}\label{eq:b5} &&G_t-c\nu e^{r(T-t)}+\alpha mG_m+\frac{1}{2} \beta^2 (G_m^{2}+G_{mm})-\lambda[(\beta_1^{-1}+\beta_2^{-1})\nu e^{r(T-t)}+1] \nonumber\\ && -\frac{1}{2}\frac{(m+a-r+\sigma\beta\rho_1G_m)^2}{\sigma^2}+\lambda[(\beta_1^{-1}+\beta_2^{-1})\nu e^{r(T-t)}+1] \nonumber\\ && \times M_Y(\frac{\beta_1\beta_2\nu e^{r(T-t)}}{(\beta_1+\beta_2)\nu e^{r(T-t)}+\beta_1\beta_2})=0, \end{eqnarray*}$

根据方程(3.5)及终端条件$G(T,m)=0$,我们有以下定理.

${\bf定理3.1}$ (3.1)式中的$G(t,m)$具有如下表达式

$\begin{equation}\label{eq:b6} G(t,m)=K(t)m^2+J(t)m+L(t), t\in[T], \end{equation}$

其中$K(t),J(t),L(t)$是下列微分方程组

$\begin{equation}\label{eq:b7} K'(t)+2(\alpha-\frac{\beta\rho_1}{\sigma})K(t)+2\beta^2(1-\rho_1^2)K^2(t)-\frac{1}{2\sigma^2}=0, \end{equation}$
$\begin{equation}\label{eq:b8} J'(t)+(\alpha-\frac{\beta\rho_1}{\sigma})J(t)-\frac{2(a-r)\beta\rho_1}{\sigma}K(t)+2\beta^2(1-\rho_1^2)K(t)J(t)-\frac{a-r}{\sigma^2}=0, \end{equation}$
$\begin{equation}\label{eq:b9} L'(t)+M(t)-\frac{(a-r)\beta\rho_1}{\sigma}J(t)+\frac{1}{2}\beta^2(1-\rho_1^2)J^2(t)+\beta^2K(t)=0 \end{equation}$

的解,其终端条件为

$\begin{equation}\label{eq:b10} K(T)=0, J(T)=0, L(T)=0, \end{equation}$

且方程(3.9)中的$M(t)$为

$\begin{eqnarray*} M(t)&=&-c\nu e^{r(T-t)}-\lambda[(\beta_1^{-1}+\beta_2^{-1})\nu e^{r(T-t)}+1]-\frac{(a-r)^2}{2\sigma^2} \nonumber\\ && +\lambda[(\beta_1^{-1}+\beta_2^{-1})\nu e^{r(T-t)}+1]M_Y(\frac{\beta_1\beta_2\nu e^{r(T-t)}}{(\beta_1+\beta_2)\nu e^{r(T-t)}+\beta_1\beta_2}). \end{eqnarray*}$

${\bf证}$ 将(3.6)式代入微分方程(3.5),得到

$\begin{eqnarray*} &&K'(t)m^2+J'(t)m+L'(t)-c\nu e^{r(T-t)}+\alpha m(2K(t)m+J(t)) \nonumber\\ && +\frac{1}{2}\beta^2[4K^2(t)m^2+4K(t)J(t)m+J^2(t)+2K(t)] \nonumber\\ && -\lambda[(\beta_1^{-1}+\beta_2^{-1})\nu e^{r(T-t)}+1]-\frac{1}{2\sigma^2}[m+a-r+\sigma\beta\rho_1(2K(t)m+J(t))]^2 \nonumber\\ && +\lambda[(\beta_1^{-1}+\beta_2^{-1})\nu e^{r(T-t)}+1]M_Y(\frac{\beta_1\beta_2\nu e^{r(T-t)}}{(\beta_1+\beta_2)\nu e^{r(T-t)}+\beta_1\beta_2})=0. \end{eqnarray*}$

将上式根据$m$的幂次方重新整理,得到

$\begin{eqnarray*} &&[K'(t)+2(\alpha-\frac{\beta\rho_1}{\sigma})K(t)+2\beta^2(1-\rho_1^2)K^2(t)-\frac{1}{2\sigma^2}]m^2 \nonumber\\ && +[J'(t) +(\alpha-\frac{\beta\rho_1}{\sigma})J(t) -\frac{2(a-r)\beta\rho_1}{\sigma}K(t)+2\beta^2(1-\rho_1^2)K(t)J(t)-\frac{a-r}{\sigma^2}]m \nonumber\\ &&+[L'(t)+M(t) -\frac{(a-r)\beta\rho_1}{\sigma}J(t)+\frac{1}{2}\beta^2(1-\rho_1^2)J^2(t)+\beta^2K(t)]=0, \end{eqnarray*}$

其中

$\begin{eqnarray*} M(t)&=&-c\nu e^{r(T-t)}-\lambda[(\beta_1^{-1}+\beta_2^{-1})\nu e^{r(T-t)}+1]-\frac{(a-r)^2}{2\sigma^2} \nonumber\\ &&+\lambda[(\beta_1^{-1}+\beta_2^{-1})\nu e^{r(T-t)}+1]M_Y(\frac{\beta_1\beta_2\nu e^{r(T-t)}}{(\beta_1+\beta_2)\nu e^{r(T-t)}+\beta_1\beta_2}). \end{eqnarray*}$

因此,如果$K(t)\mbox{、}\,J(t)\mbox{、}\,L(t)$分别是微分方程(3.7)、(3.8)和(3.9)的解,则(3.6)式是偏微分方程(3.5)的解.终端条件(3.2)式相当于(3.10)式.

接下来,对$K(t)\mbox{、}\,J(t)\mbox{、}\,L(t)$进行求解.我们首先讨论$K(t)$的黎卡提方程的解.为了便于标注,设

$D=2\beta^2(1-\rho_1^2), B=2(\alpha-\frac{\beta\rho_1}{\sigma}), C=-\frac{1}{2\sigma^2}.$

则方程(3.7)变为

$\begin{equation}\label{eq:b11} K'(t)+DK^2(t)+BK(t)+C=0, \end{equation}$

这是一个带条件$B^2-4DC>0$的正规黎卡提方程. 使用标准方法,可以得到带有终端条件$K(T)=0$的黎卡提方程(3.11)的解

$\begin{equation}\label{eq:b12} K(t)=C_1+\frac{e^{t\sqrt{B^2-4DC}}}{\frac{D}{\sqrt{B^2-4DC}}(e^{t\sqrt{B^2-4DC}}-e^{T\sqrt{B^2-4DC}})-\frac{1}{C_1}e^{T\sqrt{B^2-4DC}}}, \end{equation}$

其中

$C_1=\frac{-B-\sqrt{B^2-4DC}}{2D}.$

利用(3.12)式,带有终端条件(3.10)式的一阶线性微分方程(3.8)的解具有以下形式

$\begin{equation}\label{eq:b13} J(t)=-e^{\int_{t}^{T}p(s){\rm d}s}\int_{t}^{T}q(s)e^{-\int_{s}^{T}p(u){\rm d}u}{\rm d}s, \end{equation}$

其中

$p(s)=2K(s)\beta^2(1-\rho_1^2)+\alpha-\frac{\beta\rho_1}{\sigma},$

且

$q(s)=\frac{a-r}{\sigma^2}+\frac{2(a-r)\beta\rho_1}{\sigma}K(s).$

再者,根据(3.9)、(3.12)和(3.13)式,积分得到

$\begin{eqnarray*}\label{eq:b14} L(t)&=&\int_{t}^{T}[M(s)-\frac{(a-r)\beta\rho_1}{\sigma}J(s)+\frac{1}{2}\beta^2(1-\rho_1^2)J^2(s)+\beta^2K(s)]{\rm d}s \nonumber\\ &=& -\frac{\nu}{r}[c+\lambda(\beta_1^{-1}+\beta_2^{-1})](e^{r(T-t)}-1)-[\frac{(a-r)^2}{2\sigma^2}+\lambda](T-t) \nonumber\\ && +\int_{t}^{T}\lambda[(\beta_1+\beta_2)\nu e^{r(T-s)}+1]M_Y(\frac{\beta_1\beta_2\nu e^{r(T-s)}}{(\beta_1+\beta_2)\nu e^{r(T-s)}+\beta_1\beta_2}){\rm d}s \nonumber\\ && -\int_{t}^{T}[\frac{(a-r)\beta\rho_1}{\sigma}J(s)-\frac{1}{2}\beta^2(1-\rho_1^2)J^2(s)-\beta^2K(s)]{\rm d}s. \end{eqnarray*}$

因此, $K(t)$、$J(t)$、$L(t)$都可以解出. 证毕.

最后,我们用下面的定理总结结果.

${\bf定理3.2}$ 优化问题(2.5)式的最优投资与再保险策略是

$\begin{eqnarray*} \left\{ \begin{array}{lll} A_t^*=\frac{(m+a-r)+\sigma\beta\rho_1(2K(t)m+J(t))}{\sigma^2\nu e^{r(T-t)}},\\[3mm] q_t^*=\frac{\beta_1\beta_2}{(\beta_1+\beta_2)\nu e^{r(T-t)}+\beta_1\beta_2},\\[3mm] u_t^*=\frac{\beta_2}{\beta_1+\beta_2}, \end{array} \right.t\in(0,T]. \end{eqnarray*}$

此外,值函数是

$V(t,x,m)=\lambda_1-\frac{\eta}{\nu}e^{-\nu xe^{r(T-t)}+G(t,m),}$

其中

$G(t,m)=K(t)m^2+J(t)m+L(t),$

K(t)、J(t)和L(t)分别在(3.12)-(3.14)式中给出,终端条件为(3.10)式.

4 扩散逼近风险模型下的最优策略和值函数

在本节中,我们假设(2.1)式中的总索赔过程采用扩散逼近模型,即

$\hat{S}_t=a_0 t-\sigma_0 B_t,$

其中$a_0=\lambda\mu$, $\sigma_0^2=\lambda\mu_2$ (这里$\mu_2=E[Y_i^2]$), $B_t$是另一个独立于$W_t^{(2)}\mbox{、}\,W_t^{(1)}$的标准布朗运动. 用$\hat{S}_t$代替(2.4)式中的$S_t$,那么新的盈余过程可以表示为

$\begin{matrix}\label{eq:c1} \left\{ \begin{array}{lll} d\hat{X}_t=[r\hat{X}_t+(m(t)+a-r)A_t+c-\lambda(\beta_1^{-1}(M_Y(\beta_1 u_t(1-q_t))-1) \\ +\beta_2^{-1}(M_Y(\beta_2 (1-u_t)(1-q_t))-1))-q_ta_0]{\rm d}t+A_t \sigma {\rm d}W_t^{(1)}+q_t\sigma_0 {\rm d}B_t,\\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad(A.,q.,u.)\in \Pi, \\ dm(t)= \alpha m(t){\rm d}t+\beta {\rm d}W_t^{(2)}, m(0)=m_0. \end{array} \right. \end{matrix} $

它的值函数是

$\begin{equation}\label{eq:c2} V(t,x,m)=\max E[u(\hat{X}_T)\mid(\hat{X}_T,m(t))=(x,m)]. \end{equation}$

那么,相应的HJB方程为

$\begin{eqnarray*}\label{eq:c3} &&\sup\limits_{A,q,u}\{V_t+[rx+(m+a-r)A+c-qa_0-\lambda(\beta_1^{-1}(M_Y(\beta_1u(1-q))-1) \nonumber\\ && +\beta_2^{-1}(M_Y(\beta_2(1-u)(1-q))-1))]V_x+\alpha mV_m+\frac{1}{2}(A^2\sigma^2+q^2\sigma_0^2)V_{xx} \nonumber\\ && +\frac{1}{2}\beta^2V_{mm}+\sigma A\beta\rho_1V_{xm}\}=0, \end{eqnarray*}$

且满足终端条件$V(T,x,m)=u(x)$.同样,值函数采用(3.1)式的形式,类似于方程(3.4)的推导,可得

$\begin{eqnarray*}\label{eq:c4} &&G_t-c\nu e^{r(T-t)}+\alpha mG_m+\frac{1}{2}\beta^2(G_m^2+G_{mm})-\lambda(\beta_1^{-1}+\beta_2^{-1})\nu e^{r(T-t)} \nonumber\\ && +\inf\limits_{A}\{[-(m+a-r)A+qa_0+\frac{1}{2}(A^2\sigma^2+q^2\sigma_0^2)\nu e^{r(T-t)}-\sigma A\beta\rho_1G_m]\nu e^{r(T-t)} \nonumber\\ && +\lambda[\beta_1^{-1}M_Y(\beta_1u(1-q))+\beta_2^{-1}M_Y(\beta_2(1-u)(1-q))]\nu e^{r(T-t)}\}=0. \end{eqnarray*}$

设

$\begin{eqnarray*} g_1(A,q,u)&=&[-(m+a-r)A+qa_0+\frac{1}{2}(A^2\sigma^2+q^2\sigma_0^2)\nu e^{r(T-t)}-\sigma A\beta\rho_1G_m]\nu e^{r(T-t)} \nonumber\\ && +\lambda[\beta_1^{-1}M_Y(\beta_1u(1-q))+\beta_2^{-1}M_Y(\beta_2(1-u)(1-q))]\nu e^{r(T-t)}. \end{eqnarray*}$

对$g_1(A,q,u)$中的$A\mbox{、}\,q\mbox{、}\,u$一阶求导并令其为0,得到

$ \left\{ \begin{array}{lll} A_2(t)=\frac{(m+a-r)+\sigma\beta\rho_1G_m}{\sigma^2\nu e^{r(T-t)}},\\[3mm] u_2(t)=\frac{\beta_2}{\beta_1+\beta_2}, \end{array} \right. $
$\begin{equation}\label{eq:c5} \lambda M_Y'(\frac{\beta_1\beta_2(1-q)}{\beta_1+\beta_2})=a_0+\sigma_0^2q \nu e^{r(T-t)}, \end{equation}$

其中$M_Y'(\frac{\beta_1\beta_2(1-q)}{\beta_1+\beta_2})=E(Ye^{\frac{\beta_1\beta_2(1-q)}{\beta_1+\beta_2}Y})$. 注意, $A_2(t)\mbox{、}\,u_2(t)$与复合泊松风险模型下的$A_t^*\mbox{、}\,u_t^*$相同.那么,我们得到如下引理.

${\bf引理4.1}$ 对$\forall t\in[0,T)$, (4.5)式存在唯一解$q_2(t)\in(0,1)$.

${\bf证}$ 设$g_1(q)=\lambda M_Y'(\frac{\beta_1\beta_2(1-q)}{\beta_1+\beta_2})$, $g_2(q)=a_0+\sigma_0^2q\nu e^{r(T-t)}$.那么,我们有

$\begin{eqnarray*} \left\{ \begin{array}{lll} g_1(0)=\lambda M_Y'(\frac{\beta_1\beta_2}{\beta_1+\beta_2}), \\[2mm] g_1(1)=\lambda E(Y)=\lambda\mu,\\[2mm] g_1'(q)=-\lambda\frac{\beta_1\beta_2}{\beta_1+\beta_2}E(Y^2e^{\frac{\beta_1\beta_2(1-q)}{\beta_1+\beta_2}Y})<0,\\[3mm] g_1^{''}(q)=\lambda(\frac{\beta_1\beta_2}{\beta_1+\beta_2})^2E(Y^3e^{\frac{\beta_1\beta_2(1-q)}{\beta_1+\beta_2}Y})>0. \end{array} \right. \end{eqnarray*}$

故$g_1(q)$是一个递减的凸函数. 而对$\forall t\in[0,T)$, $g_2(q)$是一个递增的线性函数, $g_2(0)=a_0$, $g_2(1)=a_0+\sigma_0^2\nu e^{r(T-t)}$. 由于$g_1(0)>g_2(0)=g_1(1)$, $g_1(1)<g_2(1)$, $g_1(q)$与$g_2(q)$有唯一交点$q_2(t)\in(0,1)$. 证毕.

将$(A_t^*,q_t^*,u_t^*)=(A_2(t),q_2(t),u_2(t))$代入(4.4)式中, (4.4)式变为

$\begin{eqnarray*}\label{eq:c6} &&G_t-c\nu e^{r(T-t)}+\alpha mG_m+\frac{1}{2}\beta^2(G_m^2+G_{mm})-\lambda(\beta_1^{-1}+\beta_2^{-1})\nu e^{r(T-t)} \nonumber\\ && -\frac{1}{2\sigma^2}(m+a-r+\sigma\beta\rho_1G_m)^2+q_2(t)a_0\nu e^{r(T-t)}+\frac{1}{2}q_2(t)^2\sigma_0^2\nu^2e^{2r(T-t)} \nonumber\\ && +\lambda(\beta_1^{-1}+\beta_2^{-1})M_Y(\frac{\beta_1\beta_2(1-q_2(t))}{\beta_1+\beta_2})\nu e^{r(T-t)}=0. \end{eqnarray*}$

终端条件

$\begin{equation}\label{eq:c7} G(T,m)=0. \end{equation}$

类似于定理3.1,我们有以下结果.

${\bf引理4.2}$ 带有终端条件(4.7)式的微分方程(4.6)的解是$\hat{G}(t,m)=K(t)m^2+J(t)m+\hat{L}(t)$,其中

$\begin{eqnarray*}\label{eq:c8} \hat{L}(t)&=&-\frac{\nu}{r}[c+\lambda(\beta_1^{-1}+\beta_2^{-1})](e^{r(T-t)}-1)-\frac{(a-r)^2}{2\sigma^2}(T-t) +\int_{t}^{T}[q_2(s)a_0 \nonumber\\ &&+\frac{1}{2}q_2^2(s) \sigma_0^2 \nu e^{r(T-s)}+\lambda(\beta_1^{-1}+\beta_2^{-1})M_Y(\frac{\beta_1\beta_2(1-q_2(s))}{\beta_1+\beta_2})]\nu e^{r(T-s)}{\rm d}s \nonumber\\ && -\int_{t}^{T}[\frac{(a-r)\beta\rho_1}{\sigma}J(s)-\frac{1}{2}\beta^2(1-\rho_1^2)J^2(s)-\beta^2K(s)]{\rm d}s, \end{eqnarray*}$

$K(t)\mbox{、}\,J(t)$分别在(3.12)式和(3.13)式中给出.

最后,用下面的定理总结本节的主要结果.

${\bf定理4.1}$ 优化问题(4.2)式的最优投资与再保险策略是

$\begin{eqnarray*} \left\{ \begin{array}{lll} A_t^*=\frac{(m+a-r)+\sigma\beta\rho_1(2K(t)m+J(t))}{\sigma^2\nu e^{r(T-t)}}, \\[2mm] q_t^*=q_2(t),\\[2mm] u_t^*=\frac{\beta_2}{\beta_1+\beta_2}, \end{array} \right.\forall t\in(0,T]. \end{eqnarray*}$

此外,值函数是

$V(t,x,m)=\lambda_1-\frac{\eta}{\nu}e^{-\nu xe^{r(T-t)}+\hat{G}(t,m)},$

其中

$\hat{G}(t,m)=K(t)m^2+J(t)m+\hat{L}(t),$

$K(t)\mbox{、}\,J(t)$分别在(3.12)式和(3.13)式中给出,终端条件为(3.10)式, $\hat{L}(t)$在(4.8)式中给出,终端条件为$\hat{L}(T)=0$.

${\bf注4.1}$ (i)由定理$3.2$和定理$4.1$可以发现,复合泊松模型和扩散逼近模型下的最优投资策略相同, 这与Liang等[26]的结论一致.其主要原因可能是,在指数效用最大化准则下,投资策略主要依赖于金融市场而与保险市场无关.

(ii)本文中,两家再保险公司均采用指数保费准则(公平的零效用准则), $u_t$和$1-u_t$分别表示他们的再保险比例分配,该比例与两家再保险公司各自的风险厌恶参数成正比,该结论较符合实际.

(iii)与大多数文献不同的是,本文在复合泊松模型下的最优自留水平$q_t^*$比扩散逼近模型下的更为简单.这主要是因为在求解HJB方程时,复合泊松模型下的自留水平出现在索赔额分布的矩母函数中,而指数保费原理的计算同样也导致这样的问题发生,这反而给问题的求解带来简便.

5 数值分析

在本节中,我们将提供一些数值模拟来说明我们的结果.在整个数值分析过程中,除另有说明外,基本参数固定为$r=0.1$、$a=0.2$、$\sigma=1$、$\alpha=-0.4$、$\beta=-1$、$\lambda=1$、$\nu=1$、$\beta_1=0.3$、 $\beta_2=0.5$、$\rho_1=0.4$、$T=2$、$m=0.2$.

1) 复合泊松风险模型下,最优投资和再保险策略$(A_t^*,q_t^*,u_t^*)$的数值分析如下.

(1) 模型参数$\nu$、$\beta_1$、$\beta_2$对最优再保险策略$(q_t^*,u_t^*)$的影响变化

(i) 图1给出了风险厌恶参数$\nu$对自留比例$q_t^*$的影响.从图1可以看出, $q_t^*$随着$\nu$的增大而减小, 这是因为风险厌恶参数越大表示保险公司越厌恶风险,从而会购买更多的再保险,即保险公司自留额度会越小.

图 1

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图 1   $\nu$ 对最优再保险策略$q_t^*$的影响


(ii) 图2、图3分别给出了两个再保险公司的安全载荷$\beta_1$、$\beta_2$对自留比例$q_t^*$的影响. 从图2、3可以看出, $q_t^*$ 随着$\beta_1$、$\beta_2$的增大而增大,这是因为再保险公司的安全载荷越大,再保险保费就越昂贵,从而会购买更少的再保险,即保险公司自留额度会越大.

图 2

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图 2   $\beta_1$ 对最优投资策略 $q_t^*$的影响


图 3

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图 3   $\beta_2$ 对最优投资策略 $q_t^*$的影响


(iii) 而$u_t^*$只与两个再保险公司的安全载荷有关,且与$\beta_1$呈负相关,与$\beta_2$呈正相关.

(2) 模型参数$r\mbox{、}\,\nu$对最优投资策略$A_t^*$的影响变化:

(i) 由图4知,无风险利率$r$与最优投资策略$A_t^*$呈负相关.随着利率的提高,投资于无风险资产的金额在增多,从而投资于风险资产的金额在减少.

图 4

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图 4   $r$对最优投资策略 $A_t^*$的影响


(ii) 在图5中,风险厌恶系数$\nu$与最优投资策略$A_t^*$呈负相关.一个公司对风险的厌恶程度越高,选择风险投资的数目就会越少,就会更趋向于选择无风险资产,即厌恶风险的保险公司更倾向于选择低风险甚至无风险的投资方法.

图 5

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图 5   $\nu$对最优投资策略 $A_t^*$的影响


2) 扩散逼近风险模型下,最优投资和再保险策略$(A_t^*,q_t^*,u_t^*)$的数值分析如下.

该模型下的$A_t^*$和$u_t^*$与复合泊松风险模型下相同,因此这里只对$q_t^*$进行数值分析. $q_t^*=q_2(t)$是等式$\lambda M_Y'(\frac{\beta_1\beta_2(1-q)}{\beta_1+\beta_2})=a_0+\sigma_0^2q\nu e^{r(T-t)}$ 的唯一解,其中$a_0=\lambda\mu$, $\sigma_0^2=\lambda E(Y_i^2)$.这里,我们分别假设索赔额$Y$呈参数为$\lambda_2=1\mbox{、}\,2\mbox{、}\,3$的指数分布.

表1   指数分布参数$\lambda_2$ 对自留比例$q_t^*$的影响

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在表1中,固定任一指数分布,随着时间$t$的增加,保险公司的自留比例$q_t^*$小幅度增加;固定任意时刻$t$, 随着指数分布参数$\lambda_2$的增大,意味着期望索赔额的减小,则最优自留比例$q_t^*$减小,这样才能保证期望终端财富最大化.

参考文献

Browne S.

Optimal investment policies for a firm with random risk process: exponential utility and minimizing the probability of ruin

Mathematics of Operations Research, 1995, 20(4): 937-958

DOI:10.1287/moor.20.4.937      URL     [本文引用: 3]

We consider a firm that is faced with an uncontrollable stochastic cash flow, or random risk process. There is one investment opportunity, a risky stock, and we study the optimal investment decision for such firms. There is a fundamental incompleteness in the market, in that the risk to the investor of going bankrupt cannot be eliminated under any investment strategy, since the random risk process ensures that there is always a positive probability of ruin (bankruptcy). We therefore focus on obtaining investment strategies which are optimal in the sense of minimizing the risk of ruin. In particular, we solve for the strategy that maximizes the probability of achieving a given upper wealth level before hitting a given lower level. This policy also minimizes the probability of ruin. We prove that when there is no risk-free interest rate, this policy is equivalent to the policy that maximizes utility from terminal wealth, for a fixed terminal time, when the firm has an exponential utility function. This validates a longstanding conjecture about the relation between minimizing probability of ruin and exponential utility. When there is a positive risk-free interest rate, the conjecture is shown to be false. We also solve for the optimal policy for the related objective of minimizing the expected discounted penalty paid upon ruin.

Liu C S, Yang H L.

Optimal investment for an insurer to minimize its probability of ruin

North American Actuarial Journal, 2004, 8(2): 11-31

DOI:10.1080/10920277.2004.10596134      URL     [本文引用: 1]

Liang Z B.

Optimal proportional reinsurance for controlled risk process which is perturbed by diffusion

Acta Mathematicae Applicatae Sinica (English Series), 2007, 23(3): 477-488

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Chen M, Peng X F, Guo J Y.

Optimal dividend problem with a nonlinear regular-singular stochastic control

Insurance: Mathematics and Economics, 2013, 52(3): 448-456

DOI:10.1016/j.insmatheco.2013.02.010      URL     [本文引用: 1]

Meng Q B, Li Z D, Wang M H, Zhang X.

Stochastic optimal control models for the insurance company with bankruptcy return

Applied Mathematics and Information Sciences, 2013, 7(1L): 273-282

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Li D P, Rong X M, Zhao H.

Time-consistent reinsurance-investment strategy for an insurer and a reinsurer with mean-variance criterion under the CEV model

Journal of Computational and Applied Mathematics, 2015, 283: 142-162

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Xu L, Zhang L M, Yao D J.

Optimal investment and reinsurance for an insurer under Markov-modulated financial market

Insurance: Mathematics and Economics, 2017, 74: 7-19

DOI:10.1016/j.insmatheco.2017.02.005      URL     [本文引用: 1]

Wang Y J, Rong X M, Zhao H.

Optimal investment strategies for an insurer and a reinsurer with a jump diffusion risk process under the CEV model

Journal of Computational and Applied Mathematics, 2018, 328: 414-431

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Bi J N, Cai J.

Optimal investment-reinsurance strategies with state dependent risk aversion and VaR constraints in correlated markets

Insurance: Mathematics and Economics, 2019, 85: 1-14

DOI:10.1016/j.insmatheco.2018.11.007      URL     [本文引用: 1]

Chen M, Yuen K C, Wang W Y.

Optimal reinsurance and dividends with transaction costs and taxes under thinning structure

Scandinavian Actuarial Journal, 2021, 2021(3): 198-217

DOI:10.1080/03461238.2020.1824158      URL     [本文引用: 1]

Zhou Z B, Bai Y F, Xiao H L, Chen X.

A non-zero-sum reinsurance-investment game with delay and asymmetric information

Journal of Industrial and Management Optimization, 2021, 17(2): 909-936

DOI:10.3934/jimo.2020004      URL     [本文引用: 1]

Li N, Wang W.

Optimal dividend and proportional reinsurance strategy under standard deviation premium principle

Bulletin of the Malaysian Mathematical Sciences Society, 2022, 45(2): 869-888

DOI:10.1007/s40840-021-01220-w      [本文引用: 1]

Young V R, Zariphopoulou T.

Pricing dynamic insurance risks using the principle of equivalent utility

Scandinavian Actuarial Journal, 2002, 2002(4): 246-279

DOI:10.1080/03461230110106327      URL     [本文引用: 1]

Young V R.

Equity-indexed life insurance: pricing and reserving using the principle of equivalent utility

North American Actuarial Journal, 2003, 7(1): 68-86

DOI:10.1080/10920277.2003.10596078      URL     [本文引用: 1]

Moore K S, Young V R.

Pricing equity-linked pure endowments via the principle of equivalent utility

Insurance: Mathematics and Economics, 2003, 33(3): 497-516

DOI:10.1016/S0167-6687(03)00166-5      URL     [本文引用: 1]

Musiela M, Zariphopoulou T.

An example of indifference prices under exponential preferences

Finance and Stochastics, 2004, 8(2): 229-239

DOI:10.1007/s00780-003-0112-5      URL     [本文引用: 1]

Chen T, Liu W, Tan T, et al.

Optimal reinsurance with default risk: a reinsurer's perspective

Journal of Industrial and Management Optimization, 2021, 17(5): 2971-2987

DOI:10.3934/jimo.2020103      URL     [本文引用: 1]

Liang Z B, Yuen K C.

Optimal dynamic reinsurance with dependent risks: variance premium principle

Scandinavian Actuarial Journal, 2016, 2016(1): 18-36

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Han X, Liang Z B, Young V R.

Optimal reinsurance to minimize the probability of drawdown under the mean-variance premium principle

Scandinavian Actuarial Journal, 2020, 2020(10): 879-903

DOI:10.1080/03461238.2020.1788136      URL     [本文引用: 1]

Wen C.

Pricing catastrophe reinsurance under the standard deviation premium principle

AIMS Mathematics, 2022, 7(3): 4472-4484

DOI:10.3934/math.2022249      URL     [本文引用: 1]

Chi Y C, Meng H.

Optimal reinsurance arrangements in the presence of two reinsurers

Scandinavian Actuarial Journal, 2014, 2014(5): 424-438

DOI:10.1080/03461238.2012.723638      URL     [本文引用: 1]

Chen M, Yuen K C.

Optimal dividend and reinsurance in the presence of two reinsurers

Journal of Applied Probability, 2016, 53(2): 554-571

DOI:10.1017/jpr.2016.20      URL     [本文引用: 1]

\nIn this paper the optimal dividend (subject to transaction costs) and reinsurance (with two reinsurers) problem is studied in the limit diffusion setting. It is assumed that transaction costs and taxes are required when dividends occur, and that the premiums charged by two reinsurers are calculated according to the exponential premium principle with different parameters, which makes the stochastic control problem nonlinear. The objective of the insurer is to determine the optimal reinsurance and dividend policy so as to maximize the expected discounted dividends until ruin. The problem is formulated as a mixed classical-impulse stochastic control problem. Explicit expressions for the value function and the corresponding optimal strategy are obtained. Finally, a numerical example is presented to illustrate the impact of the parameters associated with the two reinsurers' premium principle on the optimal reinsurance strategy.\n

Meng H, Zhou M, Siu T K.

Optimal reinsurance policies with two reinsurers in continuous time

Economic Modelling, 2016, 59(1): 182-195

DOI:10.1016/j.econmod.2016.07.009      URL     [本文引用: 1]

Yao D J, Fan K.

Optimal risk control and dividend strategies in the presence of two reinsurers: Variance premium principle

Journal of Industrial and Management Optimization, 2017, 14(3): 1055-1083

DOI:10.3934/jimo.2017090      URL     [本文引用: 1]

Baev A V, Bondarev B V.

On the ruin probability of an insurance company dealing in a BS-market

Theory of Probability and Mathematical Statistics, 2007, 74: 11-23

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Liang Z B, Yuen K C, Guo J Y.

Optimal proportional reinsurance and investment in a stock market with Ornstein-Uhlenbeck process

Insurance: Mathematics and Economics, 2011, 49: 207-215

DOI:10.1016/j.insmatheco.2011.04.005      URL     [本文引用: 5]

Tian Y X, Guo J Y, Sun Z Y.

Optimal mean-variance reinsurance in a financial market with stochastic rate of return

Journal of Industrial and Management Optimization, 2021, 17(4): 1887-1912

DOI:10.3934/jimo.2020051      URL     [本文引用: 1]

Li Y, Mao X R, Song Y Z, Tao J.

Optimal investment and proportional reinsurance strategy under the mean-reverting Ornstein-Uhlenbeck process and net profit condition

Journal of Industrial and Management Optimization, 2022, 18(1): 75-93

DOI:10.3934/jimo.2020143      URL     [本文引用: 1]

A C X, Gu A L, Shao Y.

Optimal reinsurance and investment strategy with delay in Heston's SV model

Journal of the Operations Research Society of China, 2021, 9(2): 245-271

DOI:10.1007/s40305-020-00331-8      [本文引用: 1]

Bai Y F, Zhou Z B, Xiao H L, et al.

A hybrid stochastic differential reinsurance and investment game with bounded memory

European Journal of Operational Research, 2021, 296(2): 717-737

DOI:10.1016/j.ejor.2021.04.046      URL     [本文引用: 1]

Liu S S, Guo W J, Tong X L.

Martingale method for optimal investment and proportional reinsurance

Applied Mathematics: A Journal of Chinese Universities, 2021, 36(1): 16-30

DOI:10.1007/s11766-021-3463-8      [本文引用: 1]

Yang H, Zhang L.

Optimal investment for insurer with jump-diffusion risk process

Insurance: Mathematics and Economics, 2005, 37: 615-634

DOI:10.1016/j.insmatheco.2005.06.009      URL     [本文引用: 2]

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