保险风险模型中的分红策略最早由De Finetti^[1]引入, 之后分红问题的讨论成为风险理论的经典问题,参见文献^{[2, 3, 4, 5]}. 在已有的研究中,保险公司对资产的观测都是连续进行的,而分红通常假定为一常数的分红率, 这些在现实中都是无法实现的,无论是资产核算还是保险公司分红通常是发生在具体的时间点上, 而这些时间点均具有随机性,为了刻画这种随机性,Albrecher等引入了"随机观察"的概念, 分别在文献^{[6, 7, 8]}中讨论了Gerber-Shiu函数,分红模型和最优策略.

本文研究MAP风险模型中的最优分红问题. MAP是成批到达马氏过程的一特例, 其被广泛应用于排队论、稳定性及通讯系统领域中随机模型的构建. 近年来,MAP也被引入风险理论的研究. Cheung和Landriault在文献 ^[9] 中引入了门槛策略依赖于MAP相过程的风险模型, 在文献^[10]中推广了同一模型下的Gerber-Shiu函数; Zhang等^[11]在存在借款利率的情况下讨论了绝对破产概率; Salah和Morales^[13]在一类广义谱负MAP风险模型的基础上讨论了 Gerber-Shiu 函数. 虽然关于MAP风险模型的研究已经有了一些优美的成果, 但目前还没有关于最优分红策略的讨论.

本文将通过MAP过程从另外一个角度来构建风险模型,用MAP的相过程描述市场环境的随机性, 用MAP的跳来刻画保险公司行为的随机性.假设保险公司的盈余过程受MAP的相过程的调制, 同时保险公司只能在MAP的跳点上可以对公司的资金进行观测,在某些跳点上, 保险公司可以进行分红. 在以上假设条件下, 本文以最大化折现期望分红总量为目标,寻找随机分红点上的最优分红策略, 并分析市场经济状况和分红机会对值函数和最优策略的影响.

$(\Omega,{\cal F},P)$是一概率空间,考虑经典风险模型 \begin{equation}\label{*} S_{t}=x+ct-\sum_{i=1}^{N_{t}}Y_{i},~~t\geq 0, \tag{2.1} \end{equation} 其中$S_{0}=x\geq 0$,$N=\{N_{t},t\geq 0\}$是泊松过程, $\{Y_{i}\}_{i\in{\Bbb N}}$是一列独立同分布的随机变量,$\{Y_{i}\}_{i\in{\Bbb N}}$和 $N=\{N_{t}\}_{\geq 0}$独立.

假设复合泊松风险模型(2.1)只能在随机离散时间点$\{Z_{k}\}_{k=0}^{\infty}$ 被观测到($Z_{k}$是第$k$次观测时间点,$Z_{0}=0$),$T_{k}=Z_{k}-Z_{k-1}$ 是第$k$次观测时间间隔. 假设$\{Z_{k}\}_{k=0}^{\infty}$是MAP过程$(M(t),J(t))$的跳点. 其中 $\{M(t),t\geq 0\}$是计数过程, $J=\{J(t):t\geq 0\}$是状态空间为$E_{J}=\{1,2,\cdots ,d\}$的不可约马氏链, $d$是一有理正整数,$\{J(t),t\geq 0\}$通常称为相过程. 本文中假定$J(t)$描述经济状态的变化,$M(t)$是分红次数的计数过程. $J$的无穷小生成元记为$Q$,$Q=D_{0}+D_{1}$,其中$D_{0}=(d_{ij})_{i,j=1,\cdots ,d}$, $D_{1}=(D_{ij})_{i,j=1,\cdots ,d}$称为MAP过程的特征,MAP过程$(M(t),J(t))$ 可记为MAP($D_{0},D_{1}$). 由$J$的常返性有 \begin{equation}\label{D} (D_{0}+D_{1})\times(1,\cdots ,1)^{T}=0.\tag{2.2} \end{equation} 利用$(M(t),J(t))$的特征,可以给出观测时间间隔的条件分布 \begin{equation}\label{P1} P(T_{n+1}>t|M(Z_{n})=m,J(Z_{n})=i)=e^{d_{ii}t}, \tag{2.3} \end{equation} %其中$q_{ii}$是$Q$的对角元素,$q_{ii}=d_{ii}$.可见$T_{n+1}$是一指数分布且其参数仅依赖于状态$J(Z_{n})$. 假设$J$正处于状态$i\in E_{J}$, MAP$D_{0},D_{1}$下一步的状态变化有两种可能. 第一类,计数过程$M(t)$增加, 相过程由$i$转移到$j$,$i,j\in E_{J}$,表示保险公司进行了分红, 其转移概率为$p_{ij}^{(1)}$ \begin{equation}\label{P2} p_{ij}^{(1)}:={\rm P}(M(Z_{n+1})=m+1,J(Z_{n+1})=j|M(Z_{n})=m, J(Z_{n})=i)=-\frac{D_{ij}}{d_{ii}}.\tag{2.4} \end{equation} 第二类,计数过程$M(t)$未增加,相过程由$i$转移到$j$,$i,j\in E_{J}$, 表示保险公司只进行了观测而没有分红,其转移概率为$p_{ij}^{(0)}$ \begin{equation}\label{P3} p_{ij}^{(0)}:={\rm P}(M(Z_{n+1})=m,J(Z_{n+1})=j|M(Z_{n})=m,J(Z_{n})=i)= \left\{\begin{array}{ll} -\frac{d_{ij}}{d_{ii}},~~& j\ne i,\\ 0,& j=i. \end{array}\right. \tag{2.5} \end{equation} 对任意的$i\in E_{J}$,$\{p_{ij}^{(k)},k=0,1,j\in E_{J}\}$满足 \begin{equation} \sum_{k=0}^{1}\sum_{j\in E_{J}}p_{ij}^{(k)}=1.\tag{2.6} \end{equation}

假设风险过程(2.1)被$J$所调制,则盈余过程为 \begin{equation}\label{model} R(t)=x+\sum_{i=1}^{d}\int_{0}^{t}I_{\{J_{s}=i\}}{\rm d}S_{i}(s),~~ t\geq 0, \tag{2.7} \end{equation} 其中$x=R(0)\geq 0$是初始资产,$I_{\{.\}}$为示性函数,$\{S_{1}(t)\},\{S_{2}(t)\}, \cdots ,\{S_{d}(t)\}$为$d$个独立的风险过程 \begin{equation}\label{modeli} S_{i}(t)=x+c_{i}t -\sum_{k=1}^{N_{i}(t)}Y_{k}^{i},~~ i\in E_{J}, \tag{2.8} \end{equation} 其中$c_{i}$为常数保费率,$\{N_{i}(t)\}$是泊松过程. 赔付$\{Y_{k}^{i},k=1,2,\cdots \}$是与$Y^i$独立同分布的连续型随机变量,$Y^{i}$的概率分布函数为 $F_{Y}^{i}(.)$,密度函数为$f_{Y}^{i}(.)$,均值为$\mu_{i}$. 另外,$\{Y_{k}^{i},k=1,2,\cdots ,i=1,$ $\cdots ,d\}$,$\{N_{i}(t),i=1,\cdots ,d\}$ 和$\{(M(t),J(t)),t\geq 0\}$均相互独立.

记 \begin{equation}\label{Y} R_{n}:=S(Z_{n})-S(Z_{n-1}),~~n=1,2,\cdots \tag{2.9} \end{equation} 为盈余过程在两个观测时间点上的增量.可证明$(R_{n},T_{n})$和$(M(Z_{n}),J(Z_{n}))$ 是条件独立的 \begin{eqnarray}\label{P4} &&P(R_{n+1}\in B,T_{n+1}\in A,M(Z_{n+1})=k,J(Z_{n+1})=j| M(Z_{n})=m,J(Z_{n})=i) \\ &=&P(R_{n+1}\in B,T_{n+1}\in A| M(Z_{n})=m,J(Z_{n})=i) \\ && \times P( M(Z_{n+1})=k,J(Z_{n+1})=j| M(Z_{n})=m,J(Z_{n})=i) \\ &=&Q_{i}(B,A)\times p_{ij}^{(k-m)}, \tag{2.10} \end{eqnarray} 其中$B\in{\cal B}({\Bbb R})$,$A\subset[0,\infty)$,$i,j\in E_{J}$,$k=m,m+1$. $Q_{i}$是$R_{n+1}$和$T_{n+1}$ 在$J(Z_{n})=i$时的条件联合分布(独立于$M(t)$和$n$).

令$x\in[0,\infty)$ 表示未分红之前的盈余. 当经济状态为$i$时,分红策略集合为$D_{i}(x)=[0,x]$. $\pi=(f_{0},f_{1},\cdots )$ ,$f_{n}:{\Bbb R}_{+}\to {\Bbb R}_{+}$是一个分红策略, 其中$f_{n}(x)\in D_{J(Z_{n})}(x)$.

受控盈余过程$(X_{n})$可表示为 \begin{equation}\label{X} X_{n}=X_{n-1}-f_{n-1}(X_{n-1})+R_{n},~~n=1,2,\cdots , \tag{2.11} \end{equation} 其中$X_{0}=x$, \begin{equation}\label{f} f_{n-1}(X_{n-1})=\left\{\begin{array}{ll}0,~~& Z_{n-1}\ \mbox{为非分红时间点,} \\ g_{n-1}(X_{n-1},J(Z_{n-1})),~~& Z_{n-1} \ \mbox{为分红时间点,} \end{array}\right. \tag{2.12} \end{equation} 其中$g_{n}:{\Bbb R}_{+}\times E_{J}\to {\Bbb R}_{+}$, $g_{n}(x,i)\in D_{i}(x)$是一个分红决策.$g_{n-1}(X_{n-1},J(Z_{n-1}))=0$表明在分红点的分红量为零. 因此在分红点保险公司需要考虑是否分红,分多少,而在非分红时间点一定不可分红.令 $$\tau :=\inf\{n\in{\Bbb N}_{0}:X_{n}<0\} $$ 是破产时间,且 \begin{equation}\label{VJ} V_{i}(x;\pi):=E_{(x,i)} \bigg[\sum_{n=0}^{\tau-1}e^{-\delta Z_{n}}f_{n}(X_{n})\bigg],~~ x\in{\Bbb R}_{+} \tag{2.13} \end{equation} 为策略$\pi=(f_{0},f_{1},\cdots )$下的期望折现分红总量. 其中$\delta>0$是折现率, $E_{(x,i)}[.]$为$X_{0}=x$,$J(0)=i$时$P$下的条件期望.

保险公司的值函数为 \begin{equation}\label{Value} V_{i}(x)=\sup_{\pi}V_{i}(x;\pi),~~x\in{\Bbb R}_{+}.\tag{2.14} \end{equation}

为了利用马氏决策过程理论(详见文献^[14])处理上述问题, 本节将构建一个辅助马氏调制模型. 令$X^{*}=(X_{n}^{*})_{n\in{\Bbb N}}$ 是一状态空间为$E^{*}=\{(k,i);k=0,1,i\in E_{J}\}$的马氏链, 其转移矩阵${\rm P}=(p_{(k,i)(k',j)})_{k,k'=0,1;i,j\in E_{J}}$, \begin{equation}\label{Tr1} p_{(k,i)(1,j)}=p_{ij}^{(1)},~~k=0,1, \tag{3.1} \end{equation} \begin{equation}\label{Tr2} p_{(k,i)(0,j)}=p_{ij}^{(0)},~~k=0,1.\tag{3.2} \end{equation} $X^{*}$描述了在随机观察时间点上的经济状态和分红机会, $k=0$表示该时间点是一个观测时间点但没有分红机会, $k=1$ 表示该时间点是一个观测时间点且有分红机会,$i,j\in E_{J}$描述经济状态. 问题(2.14)可描述为下面的平稳马氏决策模型:

$ \bullet \tilde E: = [0,\infty ) \times {E^*}, (x,(k,i)) \in \tilde E;$

$\bullet$ $A_{(k,i)}:={\Bbb R}_{+}$,$a_{(k,i)}\in A_{(k,i)}$是状态 $(k,i),k=0,1,i\in E_{J}$时的分红支付;

$ \bullet {D_{(k,i)}} \subset \tilde E \times {A_{(k,i)}};$

$ \bullet {D_{(k,i)}}(x) = [0,x],x \geqslant 0$是盈余为$x$,状态为$(k,i),k=0,1,i\in E_{J}$时的策略集;

$\bullet r_{(k,i)}(x,a_{(k,i)})=:a_{(k,i)}$是盈余为$x$, 状态为$(k,i),k=0,1,i\in E_{J}$时单阶段回报;

$\bullet Q_{i}$是(2.10)式中$R_{n}$和$T_{n}$的联合分布, 则转移核为 $Q(B\times\{(k',j)\}|(x,(k,i),a_{(k,i)}))= p_{(k,i)(k',j)}Q_{i}(x-a_{(k,i)}+R_{n}\in B)$,$i,j\in E_{J}$, $k,k'=0,1$,$B\in{\cal B}({\Bbb R})$.

在辅助马氏调制模型中,分红策略为$\tilde{\pi}=(h_{0},h_{1},\cdots )$, $h_{n}:\tilde{E}\to R_{+}$可测且$h(x,(k,j))\in D_{(k,j)}(x),k=0,1$. 受控盈余过程$(X_{n})$可表示为 \begin{equation} X_{n}=X_{n-1}-h_{n}(X_{n-1},X_{n-1}^{*})+R_{n}.\tag{3.3} \end{equation} 令$E_{(x,(k,i))}$表示$(x,(k,i))\in \tilde{E}$时概率$P$下的期望. 则期望折现分红总量为 \begin{equation} V(x,(k,i);\tilde{\pi})=E_{(x,(k,i))}\bigg[\sum_{n=0}^{\tau-1} e^{-\delta Z_{n}}h_{n}(X_{n},X_{n}^{*})\bigg],~~k=0,1, \tag{3.4} \end{equation} 其中 \begin{equation}\label{nf} h_{n}(X_{n},X_{n}^{*})=\left\{\begin{array}{ll}0,& k=0 ,\\ g_{n}(X_{n},J(Z_{n})),~~& k=1, \end{array}\right. \tag{3.5} \end{equation} $g_{n}$由(2.12)式定义. 则(2.13)式中的$V_{i}(x;\pi)$可表示为 \begin{equation}\label{nV} V_{i}(x;\pi)=\left\{\begin{array}{ll}V(x,(0,i);\tilde{\pi}),~~& k=0,\\ V(x,(1,i);\tilde{\pi}),& k=1.\end{array}\right. \tag{3.6} \end{equation}

最优化问题(2.14)可描述为 \begin{equation}\label{nValue} V_{i}(x)=\left\{\begin{array}{ll}V(x,(0,i)),~~& k=0,\\ V(x,(1,i)),& k=1, \end{array}\right.\tag{3.7} \end{equation} 其中$V(x,(k,i))=\sup\limits_{\tilde{\pi}}V(x,(k,i);\tilde{\pi}),k=0,1$ 是辅助马氏调制模型中的值函数.

令$E_{i}[.]$表示$J(0)=i$时$P$下的条件期望, $R^{i}$表示$Z_{n}=i$时盈余过程在$Z_{n}$到$Z_{n+1}$上的增量, $$x^{+}:=\max(0,x),~~ \beta_{i}=:E[e^{-\delta T^{i}}],~~ C_{i}=:E[e^{-\delta T^{i}}R^{i+}], $$ $$ C=:\max_{i}E[e^{-\delta T^{i}}R^{i+}],~~ \beta=:\max_{i}E[e^{-\delta T^{i}}], $$ $$ C'=:\min_{i}\bigg(\sum_{j}p_{ij}^{(1)}\bigg)E[e^{-\delta T^{i}}R^{i+}],~~ \beta'=:\min_{i}\bigg(\sum_{j}p_{ij}^{(1)}\bigg)P_{i}[R\geq 0]E[e^{-\delta T^{i}}].$$

定义$M:=\{v:\tilde{E}\to {\Bbb R}_{+}$可测$\}$上的算子${\cal T}_{0}$为 \[\begin{align} & {{T}_{0}}v(x,(k,i))=\underset{{{a}_{(k,i)}}\in [0,x]}{\mathop{\sup }}\,\{\sum\limits_{{k}'=0}^{1}{\sum\limits_{j\in {{E}_{J}}}{{{p}_{(k,i)({k}',j)}}}}\int_{0}^{+\infty }{\int_{{{a}_{(k,i)}}-x}^{+\infty }{{{e}^{-\delta t}}}} \\ & \times v(x-{{a}_{(k,i)}}+r,({k}',j)){{Q}_{i}}(\text{d}r,\text{d}t)\}. \\ \tag{4.1}\end{align}\]

定理 4.1  a)~ 对任意$(k,i)\in E^{*}$,函数 $b(x,(k,i)):=1+x,x\geq 0;$ $b(x,(k,i)):=0,$ $x<0$是边界函数且满足 \begin{equation}\label{T01} {\cal T}_{0}^{n}b(x,(k,i))\leq\beta_{i}^{n}b(x,(k,i))+n\beta_{i}^{n-1}C_{i}.\tag{4.2} \end{equation}

b)~ (3.7)式中的值函数满足 \begin{equation}\label{bv1} x+\frac{C'}{1-\beta'}\leq V(x;(1,i))\leq x+\frac{C}{1-\beta},~~i\in E_{J}, x\in {\Bbb R}_{+}, \tag{4.3} \end{equation} \begin{equation}\label{bv2} x\sum_{j\in E_{J}}p_{ij}^{(1)}+\frac{C'}{1-\beta'}\leq V(x;(0,i))\leq V(x;(1,i)), ~~ x\in {\Bbb R}_{+}.\tag{4.4} \end{equation}

证  对 $x>0$ \begin{eqnarray} {\cal T}_{0}b(x,(k,i))&=&\sup_{a_{(k,i)}\in[0,x]} \bigg\{\sum_{k'=0}^{1}\sum_{j\in E_{J}}p_{(k,i)(k'.j)}\int_{0}^{+\infty} \int_{a_{(k,i)}-x}^{+\infty}e^{-\delta t} \\ &&\times(1+x-a_{(k,i)}+r)Q_{i}({\rm d}r,{\rm d}t)\bigg\} \\ &=&\sup_{a_{(k,i)}\in[0,x]}\int_{0}^{+\infty}\int_{a_{(k,i)}-x}^{+\infty} e^{-\delta t}(1+x-a_{(k,i)}+r)Q_{i}({\rm d}r,{\rm d}t) \\ &\leq&\int_{0}^{+\infty}\int_{-\infty}^{+\infty}e^{-\delta t}(1+x)Q_{i} ({\rm d}r,{\rm d}t)+\int_{0}^{+\infty}\int_{0}^{+\infty} e^{-\delta t}rQ_{i}({\rm d}r,{\rm d}t) \\ &=&(1+x)E[e^{-\delta T^{i}}]+E[e^{-\delta T^{i}}R^{i+}] \\ &=&(1+x)\beta_{i}+C_{i}.\tag{4.5} \end{eqnarray} 通过迭代得证.

b)~ 将$R_{n}^{i}$换为$R_{n}^{i+}$,因为$R_{n}^{i+}\geq 0$,保险公司永远不会破产, 故最优方案是立刻支付,因此 \begin{eqnarray} V(x,(k,i))&\leq& x+\sum_{j_{1}\in E_{J}}p_{ij_{1}}^{(1)}E_{i} [e^{-\delta T_{1}}R_{1}^{+}] \\ &&+\sum_{j_{1},j_{2}\in E_{J}}p_{ij_{1}}^{(0)}p_{j_{1}j_{2}}^{(1)} E_{i}[e^{-\delta T_{1}}]E_{j_{1}}[e^{-\delta T_{2}}R_{1}^{+}] \\ &&+\sum_{j_{1},j_{2},j_{3}\in E_{J}}p_{ij_{1}}^{(0)}p_{j_{1}j_{2}}^{(0)} p_{j_{2}j_{3}}^{(1)}E_{i}[e^{-\delta T_{1}}] E_{j_{1}} [e^{-\delta T_{2}}]E_{j_{2}}[e^{-\delta T_{3}}R_{1}^{+}]+\cdots \\ &&+\sum_{k=0}^{1}\sum_{j_{1},j_{2}\in E_{J}}p_{ij_{1}}^{(k)}p_{j_{1}j_{2}}^{(1)} E_{i}[e^{-\delta T_{1}}]E_{j_{1}}[e^{-\delta T_{2}}R_{2}^{+}] \\ &&+\sum_{k=0}^{1}\sum_{j_{1},j_{2},j_{3}\in E_{J}} p_{ij_{1}}^{(k)} p_{j_{1}j_{2}}^{(0)}p_{j_{2}j_{3}}^{(1)}E_{i}[e^{-\delta T_{1}}] E_{j_{1}}[e^{-\delta T_{2}}]E_{j_{2}}[e^{-\delta T_{3}}R_{2}^{+}]+\cdots \\ &&+\cdots \\ &\leq&x+C\bigg[\sum_{j_{1}\in E_{J}}p_{ij_{1}}^{(1)}+\sum_{j_{1},j_{2}\in E_{J}}p_{ij_{1}}^{(0)}p_{j_{1}j_{2}}^{(1)} +\sum_{j_{1},j_{2},j_{3}\in E_{J}} p_{ij_{1}}^{(0)}p_{j_{1}j_{2}}^{(0)} p_{j_{2}j_{3}}^{(1)}+\cdots\bigg] \\ && +C\beta\bigg[\sum_{k=0}^{1}\sum_{j_{1},j_{2}\in E_{J}}p_{ij_{1}}^{(k)} p_{j_{1}j_{2}}^{(1)}+\sum_{k=0}^{1}\sum_{j_{1},j_{2},j_{3}\in E_{J}} p_{ij_{1}}^{(k)}p_{j_{1}j_{2}}^{(0)}p_{j_{2}j_{3}}^{(1)}\cdots \bigg] \\ &&+…\\ &\leq &x+C+\beta C+\beta^{2}C+\cdots =x+\frac{C}{1-\beta}.\tag{4.6} \end{eqnarray} 考虑策略$\tilde{\pi}=(h,h,\cdots )$,其中 \begin{equation}h=h(x,(k,i))=\left\{\begin{array}{ll}0,& k=0,\\ x^{+},~~& k=1. \end{array}\right. \tag{4.7} \end{equation} 则 \begin{eqnarray} V(x,(1,i);\tilde{\pi})&\geq&x+\sum_{j_{1}\in{E_{J}}}p_{ij_{1}}^{(1)}E_{i} [e^{-\delta T_{1}}R_{1}^{+}] \\ &&+\sum_{j_{1},j_{2}\in{E_{J}}}p_{ij_{1}}^{(1)}p_{j_{1}j_{2}}^{(1)}P_{i} (R_{1}>0)E_{i}[e^{-\delta T_{1}}]E_{j_{1}}[e^{-\delta T_{2}}R_{2}^{+}] \\ &&+ \sum_{j_{1},j_{2},j_{3}\in{E_{J}}}p_{ij_{1}}^{(1)}p_{j_{1}j_{2}}^{(1)} p_{j_{2}j_{3}}^{(1)}P_{i}(R_{1}>0) P_{j_{1}}(R_{2}>0) \\ &&\times E_{i}[e^{-\delta T_{1}}]E_{j_{1}}[e^{-\delta T_{2}}] E_{j_{2}}[e^{-\delta T_{3}}R_{3}^{+}] \\ &&+\cdots \\ &\geq&x+C'+\beta'C'+ \beta'^{2}C'+\cdots =x+\frac{C'}{1-\beta}, \tag{4.8} \end{eqnarray} \begin{eqnarray} V(x,(0,i);\tilde{\pi})&\geq&\sum_{j_{1}\in{E_{J}}}p_{ij_{1}}^{(1)}(x+E_{i} [e^{-\delta T_{1}}R_{1}^{+}]) \\ &&+\sum_{j_{1},j_{2}\in{E_{J}}}p_{ij_{1}}^{(1)}p_{j_{1}j_{2}}^{(1)}P_{i}(R_{1}>0)E_{i}[e^{-\delta T_{1}}]E_{j_{1}}[e^{-\delta T_{2}}R_{2}^{+}] \\ &&+ \sum_{j_{1},j_{2},j_{3}\in{E_{J}}}p_{ij_{1}}^{(1)} p_{j_{1}j_{2}}^{(1)}p_{j_{2}j_{3}}^{(1)}P_{i}(R_{1}>0) P_{j_{1}}(R_{2}>0) \\ &&\times E_{i}[e^{-\delta T_{1}}]E_{j_{1}}[e^{-\delta T_{2}}]E_{j_{2}} [e^{-\delta T_{3}}R_{3}^{+}] \\ &&+\cdots\\ &\geq& x\sum_{j\in E_{J}}p_{ij}^{(1)}+\frac{C'}{1-\beta'}. \tag{4.9} \end{eqnarray} 证毕.

令$M_{b}:=\{v\in M:$存在$c>0$有$v\leq cb \}$,显然$V(.,(k,i))\in M_{b}$.

定理 4.2  $\{V(.,(k,i)),k=1,2,i\in E_{J}\}$满足贝尔曼方程 \begin{eqnarray} \label{Bellman} V(x,(k,i))&=&\max_{a_{(k,i)}\in [0,x]}\bigg\{a_{(k,i)} +\sum_{k'=0,1}\sum_{j\in{E_{J}}}p_{(k,i)(k',j)} \\ &&\times \int_{0}^{+\infty}\int_{a_{(k,i)}}^{+\infty} e^{-\delta t}V(x-a_{(k,i)}+r,(k',j))Q_{i}({\rm d}r,{\rm d}t)\bigg\}, \\ && x\in{\Bbb R}_{+},i\in E_{J},k=0,1.\tag{4.10} \end{eqnarray} $\{V(.,(k,i)),k=1,2,i\in{E_{J}}\}$存在,最优策略为 \begin{equation} h_{i}^{*}=h^{*}(.,(k,i))=\left\{\begin{array}{ll}0,&k=0,\\ g^{*}(.,i),~~& k=1, \end{array}\right. \tag{4.11} \end{equation} 其中$g^{*}(.,i)$是在经济状态为$i$时分红支付点上的最优策略.

证  由定理4.1 a) $\lim\limits_{n\to\infty}{\cal T}_{0}^{n}b=0$, 又 $D_{i}(x)=[0,x]$紧,映射$x\to D_{i}(x)$连续, $(x,a_{(k,i)})\to r_{(k,i)}(x,a_{(k,i)})=a_{(k,i)}$连续,$v\in M_{b}$连续,故 $$(x,a_{(k,i)})\to \int_{0}^{+\infty}\int_{a_{(k,i)}-x}^{+\infty} e^{-\delta t}v(x-a_{(k,i)}+r,(k',j))Q_{i}({\rm d}r,{\rm d}t)$$ 连续. 由文献[14,定理7.2.1]得证.

下面的两个定理给出了辅助马氏调制模型中值函数的基本性质.

定理 4.3  值函数$\{V(.,(k,i)),k=1,2,i\in{ E_{J}}\}$有下面的性质: a)~ $V(.,(k,i))$是增函数且$V(x,(1,i))-V(y,(1,i))\geq x-y,x\geq y\geq 0$.

b)~ 对$x\geq 0 $,$g^{*}(x-g^{*}(x,i),i)=0$ 且 \begin{equation}\label{f*1} V(x,(1,i))-g^{*}(x,i)=V(x-g^{*}(x,i),(1,i)).\tag{4.12} \end{equation}

证  a)~ 令 \begin{equation}\label{G} G(x;(1,i),(k',j))=\int_{0}^{+\infty}\int_{a_{(k,i)}-x}e^{-\delta t} V(x+r,(k',j))Q_{i}({\rm d}r,{\rm d}t),x\in {\Bbb R}_{+}, \tag{4.13} \end{equation} 对$k=1$,贝尔曼方程为 \begin{equation}\label{Bellman2} V(x,(1,i))=\sup_{a_{(1,i)}\in[0,x]}\bigg\{a_{(1,i)}+\sum_{k=0}^{1} \sum_{j\in E_{J}}p_{(1,i)(k,j)}G(x-a_{(1,i)};(1,i),(k,j))\bigg\}, \tag{4.14} \end{equation} 显然$V(.,(k,i))$是增函数. 令$0\leq y\leq x$, \begin{eqnarray} V(x,(1,i))&=&\sup_{a_{(1,i)}\in[0,x]}\bigg\{a_{(1,i)}+ \sum_{k=0}^{1}\sum_{j\in E_{J}}p_{(1,i)(k,j)}G(x-a_{(1,i)};(1,i),(k,j))\bigg\} \\ &\geq&\sup_{a_{(1,i)}\in[0,y]}\bigg\{x-y+a_{(1,i)}+ \sum_{k=0}^{1}\sum_{j\in E_{J}}p_{(1,i)(k,j)}G(y-a_{(1,i)};(1,i),(k,j))\bigg\} \\ &=&(x-y)+\sup_{a_{(1,i)}\in[0,y]}\bigg\{a_{(1,i)}+ \sum_{k=0}^{1}\sum_{j\in E_{J}}p_{(1,i)(k,j)}G(y-a_{(1,i)};(1,i),(k,j))\bigg\} \\ &=&(x-y)+V(y,(1,i)).\tag{4.15} \end{eqnarray}

b)~ 由贝尔曼方程 \begin{equation} V(x,(1,i))=g^{*}(x,i)+\sum_{k=0}^{1}\sum_{j\in E_{J}}p_{(1,i)(k,j)} G(x-g^{*}(x,i);(1,i),(k,j)), \tag{4.16} \end{equation} \begin{eqnarray} V(x-g^{*}(x,i),(1,i))&=&\sup_{a_{(1,i)\in[0,x-g^{*}(x,i)]}} \bigg\{a_{(1,i)} +\sum_{k=0}^{1}\sum_{j\in E_{J}}p_{(1,i)(k,j)} \\ &&\times G(x-g^{*}(x,i)-a_{(1,i)};(1,i),(k,j))\bigg\} \\ &\geq&\sum_{k=0}^{1}\sum_{j\in E_{J}}p_{(1,i)(k,j)} G(x-g^{*}(x,i);(1,i),(k,j)) \\ &=&V(x,(1,i))-g^{*}(x,i).\tag{4.17} \end{eqnarray} 由a)得 $V(x,(1,i))-V(x-g^{*}(x,i),(1,i))\geq g^{*}(x,i), $ 因此 $V(x-g^{*}(x,i),(1,i))+g^{*}(x,i)=V(x,(1,i)), $ 则 $$V(x-g^{*}(x,i),(1,i))=\sum_{k=0}^{1}\sum_{j\in E_{J}}p_{(1,i)(k,j)} G(x-g^{*}(x,i);(1,i),(k,j)).$$ 比较(4.14)式得$g^{*}(x-g^{*}(x,i),i)=0$. 若$g^{*}(x-g^{*}(x,i),i)>0$,由定义得 \begin{eqnarray} V(x-g^{*}(x,i),(1,i))&=&g^{*}(x-ug^{*}(x,i),i)+\sum_{k=0}^{1}\sum_{j\in E_{J}}p_{(1,i)(k,j)} \\ &&\times G(x-g^{*}(x,i)-g^{*}(x-g^{*}(x,i),i);(1,i),(k,j)).\tag{4.18} \end{eqnarray} 由假设在状态$(x,(1,i))$下支付$g^{*}(x,i)+g^{*}(x-g^{*}(x,i),i)$是最合理的操作,则我们得到 \begin{eqnarray} &&g^{*}(x,i)+ g^{*}(x-g^{*}(x,i),i) \\ &&+\sum_{k=0}^{1}\sum_{j\in E_{J}}G(x-g^{*}(x,i)-g^{*}(x-g^{*}(x,i),i);(1,i),(k,j)) \\ &=&g^{*}(x,i)+V(x-g^{*}(x,i),(1,i)) \\ &=&V(x,(1,i))=g^{*}(x,i)+\sum_{k=0}^{1}\sum_{j\in E_{J}}G(x-g^{*}(x,i);(1,i)(k,j)), \tag{4.19} \end{eqnarray} 故$g^{*}(x,i)+ g^{*}(x-g^{*}(x,i),i)$是 (4.14)式的最大值点,与$g^{*}(x,i)$为最大值点矛盾, 故$g^{*}(x-g^{*}(x,i),i)=0$.

本节证明最优策略是band策略.

定理 5.1  令$\xi_{i}=\sup\{x\in{\Bbb R}_{+}|g^{*}(x,i)=0\}$. 则对任意$x>\xi_{i}$, 有$\xi_{i} ＜\infty$且$g^{*}(x,i)=x-\xi_{i}$.

证  当$x>0$,$g^{*}(x,i)=0$,由定理4.1可得上界 \begin{eqnarray} V(x,(1,i))&=&\sum_{k=0}^{1}\sum_{j\in E_{J}}p_{(1,i)(k,j)}G(x-g^{*}(x,i);(1,i),(k,j)) \\ &=&\sum_{k=0}^{1}\sum_{j\in E_{J}}p_{(1,i)(k,j)} E[e^{-\delta T^{i}}V(x+R^{i},(k,j))I_{\{R^{i}\geq -x\}}] \\ &\leq&\sum_{k=0}^{1}\sum_{j\in E_{J}}p_{(1,i)(k,j)} E\bigg[e^{-\delta T^{i}}(x+R^{i}+\frac{c}{1-\beta_{i}})I_{\{R^{i}\geq -x\}}\bigg] \\ &\leq&x\beta_{i}+C+\frac{C}{1-\beta_{i}}.\tag{5.1} \end{eqnarray} 同理 $$V(x,(1,i))>x+\frac{C'}{1-\beta'}.$$ 故 \begin{equation} x\leq \frac{C}{(1-\beta_{i})^{2}}-\frac{C'}{(1-\beta')(1-\beta_{i})}.\tag{5.2} \end{equation} 因此$\xi_{i}$是有限的.

假设$g^{*}(\xi_{i},i)>0$. 由$\xi_{i}$的定义可知存在$\varepsilon\in(0,g^{*}(\xi_{i},i))$ 使得$g^{*}(\xi_{i}-\varepsilon,i)=0$. 当盈余价值为$\xi_{i}-\varepsilon$时, $g^{*}(\xi_{i},i)-\varepsilon$是一个可行支付,可得 \begin{eqnarray} &&g^{*}(\xi_{i},i)-\varepsilon+\sum_{k=0}^{1}\sum_{j\in E_{J}} p_{(1,i)(k,j)}G(\xi_{i}-\varepsilon-g^{*}(\xi_{i},i)+\varepsilon;(1,i),(k,j)) \\ &=&g^{*}(\xi_{i},i)-\varepsilon+\sum_{k=0}^{1}\sum_{j\in E_{J}} p_{(1,i)(k,j)}G(\xi_{i}-g^{*}(\xi_{i},i);(1,i),(k,j)).\tag{5.3} \end{eqnarray} 由定理4.3得 \begin{eqnarray} &&g^{*}(\xi_{i},i)+\sum_{k=0}^{1}\sum_{j\in E_{J}}p_{(1,i)(k,j)} G(\xi_{i}-g^{*}(\xi_{i},i);(1,i),(k,j)) \\ &\geq&\varepsilon+\sum_{k=0}^{1}\sum_{j\in E_{J}}p_{(1,i)(k,j)} G(\xi_{i}-\varepsilon;(1,i),(k,j)), \tag{5.4} \end{eqnarray} 则 \begin{eqnarray} &&g^{*}(\xi_{i},i)-\varepsilon+\sum_{k=0}^{1}\sum_{j\in E_{J}}p_{(1,i)(k,j)} G(\xi_{i}-\varepsilon-g^{*}(\xi_{i},i)+\varepsilon;(1,i),(k,j)) \\ &\geq&\varepsilon+\sum_{k=0}^{1}\sum_{j\in E_{J}}p_{(1,i)(k,j)} G(\xi_{i}-\varepsilon;(1,i),(k,j)). \tag{5.5} \end{eqnarray} 故在状态$(\xi_{i}-\varepsilon,(1,i))\in \tilde{E}$,$0$是最优支付, $V(\xi_{i},(1,i))-\varepsilon=V(\xi_{i}-\varepsilon,(1,i))$表明 $g^{*}(\xi_{i}-\varepsilon,i)\geq g^{*}(\xi_{i},i)-\varepsilon>0$, 矛盾,故$g^{*}(\xi,i)=0.$

令$x\geq \xi_{i}$. 由定理4.3可知$g^{*}(x-g^{*}(x,i),i)=0$, 由$\xi_{i}$的定义可知$g^{*}(x,i)\geq x-\xi_{i}$. 由于$x-g^{*}(x,i)\leq \xi_{i}\leq x$,在状态$(\xi_{i},(1,i))$支付$g^{*}(x,i)-(x-\xi_{i})$是允许的,因此 \begin{eqnarray} \label{V*} V(\xi_{i},(1,i))&\geq & g^{*}(x,i)-(x-\xi_{i})+ \sum_{k=0}^{1}\sum_{j\in E_{J}}p_{(1,i)(k,j)}G(x-g^{*}(x,i),(k,j)) \\ &=&V(x,(1,i))-(x-\xi)\geq V(x,(1,i)), \tag{5.6} \end{eqnarray} 则 \begin{equation} 0=g^{*}(\xi_{i},i)\geq g^{*}(x,i)-(x-\xi_{i}).\tag{5.7} \end{equation} 又由于$0\leq g^{*}(x,i)-(x-\xi_{i})$,故$g^{*}(x,i)=x-\xi_{i}.$

定理 5.2  $(h_{i_{1}}^{*},h_{i_{2}}^{*},\cdots )$是平稳最优策略, \begin{equation}h_{i}^{*}=h^{*}(x,(k,i))= \left\{\begin{array}{ll}0,&k=0,\\ g^{*}(x,i),~~& k=1, \end{array}\right. \tag{5.8} \end{equation} 且$g^{*}(x,i)$是band策略.

证  由定理5.1对$x>\xi_{i}$,$g^{*}(x,i)=x-\xi_{i}$.令$0\leq y ＜x\leq \xi_{i}$, \begin{eqnarray} V(x,(1,i))&=&\sup_{a_{(1,i)}\in[0,x]} \bigg\{a_{(1,i)}+\sum_{k=0}^{1}\sum_{j\in E_{J}}p_{(1,i)(k,j)}G(x-a_{(1,i)};(1,i),(k,j)) \bigg\} \\ &\geq& x-y+ \sup_{a_{(1,i)}\in[0,y]} \bigg\{a_{(1,i)}+\sum_{k=0}^{1}\sum_{j\in E_{J}}p_{(1,i)(k,j)}G(y-a_{(1,i)};(1,i),(k,j))\bigg\}. \tag{5.9}\end{eqnarray} 令$g^{*}(x',i)=\sup\limits_{0\leq x\leq \xi_{i}}g^{*}(x,i)$. 由于$g^{*}(x,i)$上半连续, 故最大值可取到. 若$g^{*}(x',i)=0$,门槛策略是最优策略. 假设$g^{*}(x',i)>0$, 则在$[x'-g^{*}(x',i),x')$上$g^{*}(x,i)=x-x'+g^{*}(x',i)$最优,在余集 $[0,\xi_{i}] \diagdown[x'-g^{*}(x',i),x']$上寻找下一个最大值点, 重复这一操作从而构建band策略.

定理5.2说明最优策略是平稳的且包含两部分,如果观测时间点不是分红点, 最优策略是分红量为$0$,如果观测时间点是分红点,最优策略是band策略, 且在不同的经济环境下,band策略有不同的波.

定理 5.3  对$i\in E_{J}$,当最大值$g^{*}(x,i)$唯一时,在分红时间点最优策略是门槛策略.