随机利率下带注资的对偶模型最优分红问题

邓丽; 郑华; 彭小飞

doi:10.6054/j.jscnun.2020033

随机利率下带注资的对偶模型最优分红问题

邓丽¹,
郑华^1, ,,
彭小飞²

1.
韶关学院数学与统计学院, 韶关 512005
2.
华南师范大学数学科学学院, 广州 510631

基金项目:

国家自然科学基金项目 11801097

广东省基础研究及应用研究重点项目 2018KZDXM065

广东省自然科学基金项目 2018A0303130274

韶关学院科研项目 SY2016KJ18

详细信息

通讯作者:
郑华, 副教授, Email:hzhengjlfdu@gmail.com

中图分类号: O211.6
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- 文章访问数: 1940
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出版历程
- 收稿日期: 2019-09-18
- 网络出版日期: 2021-03-21
- 刊出日期: 2020-04-24

The Optimal Dividend Problem in Dual Model with Capital Injections by Stochastic Interest Rates

1.
School of Mathematics and Statistics, Shaoguan University, Shaoguan 512005, China
2.
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

摘要

摘要: 在红利有界的条件下, 讨论了复合二项对偶模型中带比例交易费再注资且分红贴现利率随机变化的最优分红问题; 运用压缩映射不动点原理证明了该最优分红问题的最优值函数是一个离散的HJB方程的唯一解, 得到了最优分红策略和最优值函数的计算方法; 根据分红策略的一些性质, 得到了该最优值函数的可无限逼近的上界和下界，并采用了Bellman递归算法得到最优值函数和最优分红策略的数值解，从而得到最优分红算法.数值实例结果表明:该最优分红策略是有效的.这为公司的决策者在兼顾公司正常运营和股东利益而进行红利决策时提供了理论依据.
- 最优分红 /
- 随机利率 /
- 再注资 /
- HJB方程
Abstract: Under the dividend bounded, the problem of optimal dividend payment in compound binomial dual model with proportional transaction cost capital injections and stochastic interest rates was discussed. It was proved with the fixed-point principle of contraction mapping that the optimal value function of this optimal dividend problem was the unique solution to a discrete Hamilton-Jacobi-Bellman(HJB) equation. The algorithm was obtained for the optimal dividend strategy and the optimal value function. According to some properties of the dividend strategy, the upper and lower bounds of the optimal value function were derived, and the numerical solutions to the optimal value function and the optimal dividend strategy was obtained with the Bellman recursive algorithm, and the optimal dividend algorithm was obtained. The numerical result shows that the optimal dividend strategy is effective. A theoretical basis for the decision-maker to make dividend policy in consideration of the normal operation of the company and interests of shareholders is provided.
- optimal dividend /
- stochastic interest rates /
- capital injections /
- HJB equation

HTML全文

随着公司股份制改革, 为了保证公司现金流的充足和运营安全, 公司在发行股票筹集资金时需考虑回馈作为投资者的股东, 即分红.分红问题中, 讨论得较多的有经典风险模型和风险对偶模型^[1-6], 其中风险对偶模型可描述为^[1]:

$U\left( t \right) = u - ct + S\left( t \right)\;\;\;\;\left( {t \in {\mathbb{N}_ + },u \in \mathbb{N}} \right),$

(1)

其中, U(0)=u表示公司的初始资金, 正整数c表示在单位时间(t-1, t]内的支出, S(t)表示直到t时刻的总收益.

风险对偶模型中研究得较多的分红策略有:Barrier策略^[1]和Threshold策略^[2], 例如:运用Laplace变换方法讨论了复合Poisson对偶模型的最优分红Barrier的确定方法^[1]; 利用2个Integro积分方程和Laplace变换给出了最优分红Threshold的计算方法^[2].显然, 在最优分红问题中, 既提高股东收益又降低公司风险的方案应当同时考虑分红与再注资^[7-13], 例如:在离散经典风险模型中证明了最优值函数是一个Hamilton-Jacobi-Bellman(HJB)方程的唯一解, 指出再注资后最优分红策略是Barrier策略^[7]; 在对偶模型中证明了带注资的最优分红策略为Barrier策略^[8]; 讨论了带比例和固定交易费的再注资的最优分红问题, 并通过数值实例说明分红边界随交易费比例的增加而上升^[9]; 在复合二项风险模型中证明了最优值函数是一个HJB方程的唯一解, 并验证了最优控制策略是双Barrier策略^[10].为了更贴合实际, 考虑分红贴现利率的变化具有随机性^[14-15], 例如:在随机利率下讨论了离散风险模型中具有延迟索赔的最优分红问题, 得到了最优策略的一个高效算法^[14].

本文在红利有界的条件下, 研究复合二项对偶模型中带比例交易费再注资且分红贴现利率随机变化的最优分红问题; 运用压缩映射不动点原理证明了该最优分红问题的最优值函数是一个离散HJB方程的唯一解, 得到了最优分红策略和最优值函数的计算方法; 为了能在实际运用中计算最优红利值, 根据分红策略的一些性质得到了该最优值函数的可无限逼近的上界和下界; 最后给出数值实例来验证本文所给的最优分红策略的有效性.

1. 预备知识

文中用到的相关记号、符号如下:

(ⅰ) $\mathbb{N}$ ={0, 1, 2, …}; $\mathbb{N}$ ₊={1, 2, …}; $\mathbb{N}$ _k={0, 1, 2, …, k} (k∈ $\mathbb{N}$ ).

(ⅱ) c, C, m, x∈ $\mathbb{N}$ ₊; i, j=1, 2, …, m.

(ⅲ) $\mathcal{F}$ _t是σ代数, 包含了t时刻及之前的所有信息.

(ⅳ) a₁ ∨ a₂=max{a₁, a₂}, a₁ ∧ a₂=min{a₁, a₂}.

假设任意单位时间(t-1, t] (t∈ $\mathbb{N}$ ₊)内至多有一次收入, 在t的前一瞬时结算.用ε_t=1表示有一次收入，收入量为X_t∈ $\mathbb{N}$ ₊; ε_t=0表示无收入.序列{X_t}和{ε_t}分别为独立同分布的随机变量, 其中{ε_t}具有概率Pr(ε_t=1)=p (0 < p < 1), Pr(ε_t=0)=q=1-p, {X_t}具有概率函数f(x)=Pr(X_t=x)和分布函数 $F(x) = \sum\limits_{l = 1}^x f (l)$ , 且与{ε_t}相互独立.则直到时刻t的总收益S(t)是复合二项序列^[16]:

$S\left( t \right) = \left\{ \begin{array}{l} 0\;\;\;\;\left( {t = 0} \right),\\ {X_1}{\varepsilon _1} + {X_2}{\varepsilon _2} + \cdots + {X_t}{\varepsilon _t}\;\;\;\;\left( {t \ge 1} \right). \end{array} \right.$

再假设(t-1, t]时间段的利率{R_t, t∈ $\mathbb{N}$ ₊}是有限状态空间{r₁, r₂, …, r_m}的关于 $\mathcal{F}$ _t可测的齐次Markov链, 一步转移概率阵为P=(p_ij)_{i, j=1}^m, 其中p_ij=Pr(R_t+1=r_j|R_t=r_i).令v_i (0 < v_i < 1)为第i时间段内的贴现因子, v=max{v₁, v₂, …, v_m}, 则v_i=1/(1+r_i).

在模型(1)中引入分红策略.假设在0时刻不考虑分红, 分别用d_t和z_t表示t(t∈ $\mathbb{N}$ )时刻的分红和再注资, 记d₀=0.在任何时刻t的分红策略满足以下4个条件称为可行的策略:(1)d_t取整数且有上界C; (2)d_t和z_t关于 $\mathcal{F}$ _t可测; (3)盈余u不大于c时不分红, 并由股东注入相应的资金c-u(存在交易费), 使盈余能够快速恢复到c; (4)盈余u不小于c时不注资, 分红由超出c的部分承担.

由于模型(1)具有Markov性质, 所以只需讨论依赖于各时刻盈余的可行策略, 这类策略是关于盈余的函数, 组成的集合用Λ表示.为方便讨论, 用φ_i(u)表示盈余为u、利率状态为r_i时的分红量, 则任意时刻t (t∈ $\mathbb{N}$ ₊)的利率R_t=r_j时的分红量为φ_j(U(t-1)-c+X_tε_t).故初始利率为R₀=r_i的累积分红折现均值函数(以下简称值函数)^[7]为

$\begin{array}{*{20}{c}} {{V_i}\left( {u;{\varphi _i}} \right) = E\left[ {\sum\limits_{k = 1}^\infty {{d_k}} \left( {\prod\limits_{t = 1}^k {\frac{1}{{1 + {R_t}}}} } \right) - } \right.}\\ {\left. {\sum\limits_{k = 0}^\infty {{z_k}} \left( {\prod\limits_{t = 1}^k {\frac{1}{{1 + {R_t}}}} } \right)\left| {{R_0} = {r_i}} \right.} \right],} \end{array}$

(2)

记 $\prod\limits_{t = 1}^0 {\frac{1}{{1 + {R_t}}}} = 1$ .

优化目标是找到最优值函数

$V_i^ * \left( {u;{\varphi _i}} \right) = \mathop {\sup }\limits_{{\varphi _i} \in \mathit{\Lambda }} {V_i}\left( {u;{\varphi _i}} \right)$

和对应的最优分红策略, 记为φ_i^*(u)=argmax V_i(u, φ_i).为方便讨论, V_i(u; φ_i)记为V_i(u).

2. 最优分红策略和最优值函数

对任意的φ_i(u)∈Λ, 根据全期望公式^[17], 带比例交易费再注资的值函数V_i(u)满足:

${V_i}\left( u \right) = \left\{ \begin{array}{l} (1 + \beta )(u - c) + {V_i}(c)\;\;\;\;(0 \le u < c),\\ q\sum\limits_{j = 1}^m {{p_{ij}}} {v_j}\left[ {{V_j}\left( {u - c - {\varphi _j}(u - c)} \right) + {\varphi _j}(u - c)} \right] + \\ \;\;\;p\sum\limits_{j = 1}^m {{p_{ij}}} {v_j}\sum\limits_{x = 1}^\infty {\left[ {{V_j}\left( {u - c + x - {\varphi _j}(u - c + x)} \right)} \right.} \\ \;\;\;\left. {{\varphi _j}(u - c + x)} \right]f(x)\quad (u \ge c), \end{array} \right.$

(3)

其中β(β≥0)为交易费的比例(常量).

定理1 对任意的φ_i(u)∈Λ, 最优值函数V_i^*(u)满足如下HJB方程:

$\begin{array}{*{20}{l}} V_i^ * \left( u \right) = \hfill \\ \left\{ \begin{array}{*{20}{l}} (1 + \beta )(u - c) + V_i^ * (c)\;\;\;\;(0 \leqslant u < c), \hfill \\ q\sum\limits_{j = 1}^m {{p_{ij}}} {v_j}\left[ {\mathop {\max }\limits_{a \in {\mathbb{N}_{\left( {C \wedge \left( {u - 2c} \right)} \right) \vee 0}}} \left( {{V_j}\left( {u - c - a} \right) + a} \right)} \right] + \hfill \\ \;\;\;p\sum\limits_{j = 1}^m {{p_{ij}}} {v_j}\sum\limits_{x = 1}^\infty {\left[ {\mathop {\max }\limits_{a \in {\mathbb{N}_{\left( {C \wedge \left( {u - 2c + x} \right)} \right) \vee 0}}} \left( {{V_j}(u - c + x - a) + a} \right)f(x)} \right]} \hfill \\ \;\;\;\left( {u \geqslant c} \right). \hfill \\ \end{array} \right. \hfill \\ \end{array}$

(4)

证明对式(3)取最优值得:

$\begin{array}{l} V_i^ * \left( u \right) = \\ \left\{ \begin{array}{l} \mathop {\sup }\limits_{{\varphi _i}\left( u \right) \in \mathit{\Lambda }} \left\{ {\left( {1 + \beta } \right)\left( {u - c} \right) + {V_i}\left( c \right)} \right\}\;\;\;\;\left( {0 \le u < c} \right),\\ \mathop {\sup }\limits_{{\varphi _i}\left( u \right) \in \mathit{\Lambda }} \left\{ {q\sum\limits_{j = 1}^m {{p_{ij}}} {v_j}\left[ {{V_j}\left( {u - c - {\varphi _j}\left( {u - c} \right)} \right) + {\varphi _j}\left( {u - c} \right)} \right] + } \right.\\ \;\;\;\;p\sum\limits_{j = 1}^m {{p_{ij}}} {v_j}\sum\limits_{x = 1}^\infty {\left[ {{V_j}\left( {u - c + x - {\varphi _j}(u - c + x)} \right) + } \right.} \\ \;\;\;\;\left. {\left. {{\varphi _j}\left( {u - c + x} \right)} \right]f\left( x \right)} \right\}\;\;\;\;\left( {u \ge c} \right). \end{array} \right. \end{array}$

(5)

显然式(5)与式(4)等价.证毕.

定义1^[16] 记H表示所有m维有界实序列组成的集合.对H中任意两点X=(X_i(u))和Y=(Y_i(u))(u∈ $\mathbb{N}$ , i=1, 2, …, m), 称

$d\left( {\mathit{\boldsymbol{X}},\mathit{\boldsymbol{Y}}} \right) = \left\| {\mathit{\boldsymbol{X}} - \mathit{\boldsymbol{Y}}} \right\| = \mathop {\sup }\limits_{u,i} \left| {{X_i}\left( u \right) - {Y_i}\left( u \right)} \right|$

为X、Y的距离.显然H=(H, d)是完备度量空间, 且任意的V =(V_i(u))∈H.

对任意的V =(V_i(u))∈H (i={1, 2, …, m}), 定义

$\begin{array}{*{20}{l}} {b_{{V_i}}}\left( u \right) = \left\{ \begin{array}{*{20}{l}} 0\;\;\;\;\left( {0 \leqslant u \leqslant c} \right), \hfill \\ \arg \mathop {\max }\limits_{d \in {\mathbb{N}_{C \wedge \left( {u - c} \right)}}} \left\{ {{V_i}\left( {u - d} \right) + d} \right\}\;\;\;\;\;\left( {u > c} \right). \hfill \\ \end{array} \right. \end{array}$

又因式(4)中当0≤u < c时, V_i^*(u)的解的情况仅依赖于V_i^*(c)的解, 下面讨论u≥c时V_i^*(u)的解的情况.

记T =(T₁, T₂, …, T_m), 其中T_i (i=1, 2, …, m)为H上的m个算子, 满足:

$\begin{array}{l} {T_i}\mathit{\boldsymbol{V}} = q\sum\limits_{j = 1}^m {{p_{ij}}} {v_j}\left[ {{V_j}\left( {u - c - {b_{{V_j}}}\left( {u - c} \right)} \right) + {b_{{V_j}}}\left( {u - c} \right)} \right] + \\ \;\;\;\;\;\;\;\;p\sum\limits_{j = 1}^m {{p_{ij}}} {v_j}\sum\limits_{x = 1}^\infty {\left[ {{V_j}\left( {u - c + x - {b_{{V_j}}}\left( {u - c + x} \right)} \right) + } \right.} \\ \;\;\;\;\;\;\;\;\left. {{b_{{V_j}}}\left( {u - c + x} \right)} \right]f\left( x \right)\;\;\;\;\left( {u \ge c} \right), \end{array}$

(6)

或

$\mathit{\boldsymbol{TV}} = \left( {{T_1}\mathit{\boldsymbol{V}},{T_2}\mathit{\boldsymbol{V}}, \cdots ,{T_m}\mathit{\boldsymbol{V}}} \right),$

(7)

则式(4)中u≥c的部分等价于

${T_i}\mathit{\boldsymbol{V}} = {V_i},$

(8)

或

$\mathit{\boldsymbol{TV}} = \mathit{\boldsymbol{V}}.$

(9)

定理2 方程(9)有且仅有唯一解.

证明假设对任意X_i(u), Y_i(u)∈H, i={1, 2, …, m}, 及给定的u≥c, X_i(u-b_{X_i}(u))+b_{X_i}(u)≥Y_i(u-b_{Y_i}(u))+b_{Y_i}(u), 因为Y_i(u-b_{Y_i}(u))+b_{Y_i}(u)≥Y_i(u-b_{X_i}(u))+b_{X_i}(u), 所以

$\begin{array}{*{20}{l}} \mathop {\sup }\limits_{u \geqslant c} \left| {{X_i}\left( {u - {b_{{X_i}}}\left( u \right)} \right) + {b_{{X_i}}}\left( u \right) - \left[ {{Y_i}\left( {u - {b_{{Y_i}}}\left( u \right)} \right) + {b_{{Y_i}}}\left( u \right)} \right]} \right| \leqslant \hfill \\ \;\;\;\mathop {\sup }\limits_{u \geqslant c} \left| {{X_i}\left( {u - {b_{{X_i}}}\left( u \right)} \right) + {b_{{X_i}}}\left( u \right) - \left[ {{Y_i}\left( {u - {b_{{X_i}}}\left( u \right)} \right) + {b_{{X_i}}}\left( u \right)} \right]} \right| \leqslant \hfill \\ \;\;\;\mathop {\sup }\limits_{u \geqslant c} \left| {{X_i}\left( u \right) - {Y_i}\left( u \right)} \right| \leqslant \left\| {\mathit{\boldsymbol{X}} - \mathit{\boldsymbol{Y}}} \right\|, \hfill \\ \end{array}$

则

$\begin{gathered} \mathop {\sup }\limits_{u \geqslant c} \left| {{T_i}\mathit{\boldsymbol{X}} - {T_i}\mathit{\boldsymbol{Y}}} \right| \leqslant q\sum\limits_{j = 1}^m {{p_{ij}}} {v_j}\left\| {\mathit{\boldsymbol{X}} - \mathit{\boldsymbol{Y}}} \right\| + \hfill \\ \;\;\;\;p\sum\limits_{i = 1}^m {{p_{ij}}} {v_j}\sum\limits_{x = 1}^\infty {\left\| {\mathit{\boldsymbol{X}} - \mathit{\boldsymbol{Y}}} \right\|} f(x) \leqslant v\left\| {\mathit{\boldsymbol{X}} - \mathit{\boldsymbol{Y}}} \right\|. \hfill \\ \end{gathered}$

故对任意的u≥c, 有

$d\left( {\mathit{\boldsymbol{TX}},\mathit{\boldsymbol{TY}}} \right) \leqslant vd\left( {\mathit{\boldsymbol{X}},\mathit{\boldsymbol{Y}}} \right).$

在0 < v < 1的假设下, 算子T是压缩映射, 因此, 当u≥c时, 方程(9)有且仅有唯一解.证毕.

结论1 对任意u∈ $\mathbb{N}$ , i={1, 2, …, m}, 方程(4)有且仅有唯一解, 且该唯一解为最优值函数V_i^*(u).

定理3 对任意的φ_i(u)∈Λ, u∈ $\mathbb{N}$ , i={1, 2, …, m}, V_i(u)取最优值时当且仅当

$\begin{array}{*{20}{l}} \varphi _i^*\left( u \right) = \left\{ {\begin{array}{*{20}{l}} 0\ \ \ \ \ {(0 \leqslant u \leqslant c),} \\ {\arg \mathop {\max }\limits_{_{d \in {\mathbb{N}_{C \wedge (u - c)}}}} \left( {{V_i}(u - d) + d} \right)}&{(u > c).} \end{array}} \right. \end{array}$

(10)

证明 (1)必要性.由模型的假设知, 当0≤u≤c时, φ_i(u)=0.对任意的i={1, 2, …, m}, 若V_i(u)是最优的值函数, 则V_i(u)满足式(3)、(4).比较式(3)与式(4), 当u>c时, 有

${V_i}\left( {u - {\varphi _i}\left( u \right)} \right) + {\varphi _i}\left( u \right) = \mathop {\max }\limits_{_{d \in {\mathbb{N}_{C \wedge (u - c)}}}} \left( {{V_i}(u - d) + d} \right),$

因此式(10)成立.

(2) 充分性.因方程(4)的解存在且唯一, 故充分性成立.证毕.

3. 最优红利值的算法

采用Bellman递归算法, 计算最优值函数列(V_i^*(u))和最优分红策略φ_i^*(u) (u∈ $\mathbb{N}$ )的值^[16].即任意给定一个初始函数列(V_i⁽⁰⁾(u)), 根据下式计算(V_i^(s)(u)) (s=1, 2, …):

$\begin{array}{*{20}{l}} V_i^{(s)}(u) = \hfill \\ \left\{ \begin{array}{*{20}{l}} \left( {1 + \beta } \right)\left( {u - c} \right) + V_i^{\left( {s - 1} \right)}(c)\;\;\;\;\left( {0 \leqslant u < c} \right), \hfill \\ q\sum\limits_{j = 1}^m {{p_{ij}}} {v_j}\left[ {V_j^{(s - 1)}\left( {u - c - {b_{V_j^{\left( {s - 1} \right)}}}\left( {u - c} \right)} \right) + {b_{V_j^{\left( {s - 1} \right)}}}\left( {u - c} \right)} \right] + \hfill \\ p\sum\limits_{j = 1}^m {{p_{ij}}} {v_j}\sum\limits_{x = 1}^\infty {\left[ {V_j^{(s - 1)}\left( {u - c + x - {b_{V_j^{(s - 1)}}}(u - c + x)} \right) + } \right.} \hfill \\ \left. {{b_{V_j^{(s - 1)}}}(u - c + x)} \right]f(x)\;\;\;\;(u \geqslant c). \hfill \\ \end{array} \right. \hfill \\ \end{array}$

(11)

由不动点原理及式(8)、(11)得

$V_i^ * \left( u \right) = \mathop {\lim }\limits_{n \to \infty } V_i^{(n)}(u) = \mathop {\lim }\limits_{n \to \infty } T_i^nV_i^{(0)}(u),$

n充分大时可以用V_i⁽ⁿ⁾(u)近似V_i^*(u), 它们之间的误差估计为

$d\left( {V_i^{(n)},V_i^*} \right) \leqslant \frac{{{v^n}}}{{1 - v}}.$

(12)

当时间区间(t-1, t]内可能的收入不是有界的随机变量时, 递归式(11)中的函数序列存在无穷项之和, 不便于数值计算.在C < ∞的情形下, 对任意的n₁∈ $\mathbb{N}$ ₊, 考虑如下2个方程:

$\begin{array}{*{20}{l}} V_i^{*\left( 1 \right)}\left( u \right) = \hfill \\ \;\;\;\left\{ \begin{array}{*{20}{l}} (1 + \beta )(u - c) + V_i^{*(1)}(c)\quad (0 \leqslant u < c), \hfill \\ q\sum\limits_{j = 1}^m {{p_{ij}}} {v_j}\left[ {V_j^{*(1)}\left( {u - c - {b_{V_j^*}}(u - c)} \right) + {b_{V_j^*(1)}}(u - c)} \right] + \hfill \\ \;\;\;p\sum\limits_{j = 1}^m {{p_{ij}}} {v_j}\sum\limits_{x = 1}^{{n_1}} {\left[ {V_j^{*(1)}\left( {u - c + x - {b_{V_j^*(1)}}(u - c + x)} \right) + } \right.} \hfill \\ \left. {{b_{V_j^*(1)}}(u - c + x)} \right]f(x)\;\;\;\;\;(u \geqslant c); \hfill \\ \end{array} \right. \hfill \\ \end{array}$

(13)

$\begin{array}{*{20}{l}} V_i^{ * \left( 2 \right)}\left( u \right) = \hfill \\ \;\;\;\left\{ \begin{array}{*{20}{l}} (1 + \beta )(u - c) + V_i^{*(2)}(c)\;\;\;\;(0 \leqslant u < c), \hfill \\ q\sum\limits_{j = 1}^m {{p_{ij}}} {v_j}\left[ {V_j^{*(2)}\left( {u - c - {b_{V_j^{*(2)}}}(u - c)} \right) + {b_{V_j^{*(2)}}}(u - c)} \right] + \hfill \\ \;\;\;\;p\sum\limits_{j = 1}^m {{p_{ij}}} {v_j}\sum\limits_{x = 1}^{{n_1}} {\left[ {V_j^{*(2)}\left( {u - c + x - {b_{V_j^*(2)}}(u - c + x)} \right) + } \right.} \hfill \\ \left. {\;\;\;\;\;{b_{{V_j}*(2)}}(u - c + x)} \right]f(x) + \frac{{pvC}}{{1 - v}}\bar F\left( {{n_1}} \right)\quad (u \geqslant c). \hfill \\ \end{array} \right. \hfill \\ \end{array}$

(14)

定理4 对任意的u ∈ $\mathbb{N}$ , i={1, 2, …, m}, 有

$V_i^{*(1)}(u) \leqslant V_i^*(u) \leqslant V_i^{*(2)}(u).$

(15)

证明 (1)先证V_i^*(1)(u)≤V_i^*(u) (u≥c).

对任意的u≥c和给定的i, 记W_i⁽¹⁾(u)为方程(6)等号的右边, 则V_i^*(1)(u)≤W_i⁽¹⁾(u)≤T_iW_i⁽¹⁾(u).记W_i⁽²⁾(u)=T_iW_i⁽¹⁾(u), 则W_i⁽¹⁾(u)≤W_i⁽²⁾(u)≤T_iW_i⁽²⁾(u).依次类推, 可得一个递增的函数序列{W_i^(n₂)(u), n₂=1, 2, …}满足W_i^(n₂+1)=T_iW_i^(n₂), 有

$\mathop {\lim }\limits_{{n_2} \to \infty } W_i^{\left( {{n_2}} \right)}(u) = V_i^*(u).$

因此,

$V_i^{*(1)}(u) \leqslant V_i^*(u)\quad (u \geqslant c)$

(2) 同理可证V_i^*(u)≤V_i⁽²⁾(u) (u≥c)^[18].

(3) 由证明过程(1)、(2), 当u≥c时, 有V_i^*(1)(u)≤V_i^*(u)≤V_i^*(2)(u); 由式(4)、(13)、(14), 当0≤u < c时, 有V_i^*(1)(u)≤V_i^*(u)≤V_i^*(2)(u).证毕.

推论1 对任意的0 < v < 1, n₁, u∈ $\mathbb{N}$ , i={1, 2, …, m}, 有

$d\left( {V_i^{*(1)},V_i^{*(2)}} \right) \leqslant \frac{{pvC\bar F\left( {{n_1}} \right)}}{{(1 - v)\left( {1 - qv - pvF\left( {{n_1}} \right)} \right)}}.$

(16)

证明由式(13)、(14), 当0≤u < c时, 有

$d\left( {V_i^{*(1)}(u),V_i^{*(2)}(u)} \right) \leqslant d\left( {V_i^{*(1)}(c),V_i^{*(2)}(c)} \right).$

当u≥c时, 由定理2的证明过程，有

$\begin{gathered} d\left( {V_i^{*(1)},V_i^{*(2)}} \right) \leqslant q\sum\limits_{i = 1}^m {{p_{ij}}} {v_j}d\left( {V_i^{*(1)},V_i^{*(2)}} \right) + \hfill \\ \;\;\;\;\;p\sum\limits_{i = 1}^m {{p_{ij}}} {v_j}\sum\limits_{x = 1}^{{n_1}} d \left( {V_i^{*(1)},V_i^{*(2)}} \right)f(x) + \frac{{pvC}}{{1 - v}}\bar F\left( {{n_1}} \right) \leqslant \hfill \\ \;\;\;\;\;\left( {qv + pvF\left( {{n_1}} \right)} \right)d\left( {V_i^{*(1)},V_i^{*(2)}} \right) + \frac{{pvC}}{{1 - v}}\bar F\left( {{n_1}} \right), \hfill \\ \end{gathered}$

则式(16)成立.证毕.

当n₁充分大时, V_i^*(1)(u)和V_i^*(2)(u)能无限逼近最优值函数V_i^*(u).定义算子 $\widetilde{\boldsymbol{T}}=\left(\tilde{T}_{1}, \tilde{T}_{2}, \cdots\right., \left.\tilde{T}_{m}\right)$ , 其中 $\widetilde{\boldsymbol{T}} V_{i}^{*(1)}(u)$ 等于式(13)等号的右边.显然, $\tilde{\boldsymbol{T}}$ 也是H上的压缩映射.运用式(13)或式(14)变换迭代公式(11)后，可进行数值实例的计算.

综上可得最优红利值的算法:

第1步, 精度控制.给定一个精度要求, 根据式(12)得到迭代步数n及式(16)中n₁的值.

第2步, 迭代计算.任意给定一个初始函数列(V_i⁽⁰⁾(u)) (0≤u≤n₁(n+1)), 由式(13)或式(14)变换迭代公式(11)，然后计算(V_i^(s)(u)) (0≤u≤n₁(n-s+1)), 其中s={1, 2, …, n}, 最终得到φ_i^*(u)和V_i^*(u)的近似值.

4. 数值模拟

本节列举了一个几何分布的实例, 利用第3节提出的最优红利值的算法来计算函数φ_i^*(u)和V_i^*(u), 其中转移概率矩阵采用与文献[14]、[19]类似的方法构造.为进一步对比分析, 计算不注资时的最优分红策略和最优值函数, 满足条件:(1)分红有上界, 且任何时刻的分红都不超过该时刻的盈余; (2)当盈余为负值时公司破产.不注资时的最优值函数W_i^*(u)为^[16]:

$\begin{array}{*{20}{l}} W_i^*(u) = q\sum\limits_{j = 1}^m {{p_{ij}}} {v_j}\left( {W_j^*\left( {u - c - {b_{W_j^*}}(u - c)} \right) + {b_{W_j^*}}(u - c)} \right) + \hfill \\ \;\;\;\;\;\;\;\;p\sum\limits_{j = 1}^\infty {{p_{ij}}} {v_j}\sum\limits_{x = 1}^\infty {\left( {W_j^*\left( {u - c + x - {b_{W_j^*}}(u - c + x)} \right) + } \right.} \hfill \\ \;\;\;\;\;\;\;\;\left. {{b_{W_j^*}}(u - c + x)} \right)f(x)\;\;\;\;\;(u \geqslant 0). \hfill \\ \end{array}$

(17)

例1 假设p=0.7, c=C=5, 收入量服从均值为μ=10的几何分布, 概率函数为:

$f(x) = \frac{1}{{10}}{\left( {\frac{9}{{10}}} \right)^{x - 1}}\;\;\;\;\;(x = 1,2, \cdots ).$

再假设利率为:r₁=0.05, r₂=0.04, r₃=0.03, 转移概率矩阵为:

$\mathit{\boldsymbol{P}} = \left( {\begin{array}{*{20}{c}} {0.5}&{0.2}&{0.3} \\ {0.25}&{0.6}&{0.15} \\ {0.1}&{0.2}&{0.7} \end{array}} \right).$

表 1是当交易费比例为β=0, 0.2, 0.4, 0.6, 1.2及不注资时, 最优分红策略φ_i^*(u)对应的值.表 2是当交易费比例为β=0, 0.2, 1.2以及不注资时, 最优值函数V_i^*(u)的部分近似值(精度为10^-4).

表 1 最优分红策略φ_i^*(u) (i=1, 2, 3, C=5)

Table 1. The optimal dividend strategy φ_i^*(u) (i=1, 2, 3, C=5)

u	β=0			β=0.2			β=0.4			β=0.6			β=1.2			不注资
u	i=1	i=2	i=3	i=1	i=2	i=3	i=1	i=2	i=3	i=1	i=2	i=3	i=1	i=2	i=3	i=1	i=2	i=3
0, 1, …, 10	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
11	1	1	1	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0
12	2	2	2	2	2	2	1	0	0	0	0	0	0	0	0	0	0	0
13	3	3	3	3	3	3	2	1	1	0	0	0	0	0	0	0	0	0
14	4	4	4	4	4	4	3	2	2	0	0	0	0	0	0	0	0	0
15	5	5	5	5	5	5	4	3	3	0	0	0	0	0	0	0	0	0
16	5	5	5	5	5	5	5	4	4	1	1	1	0	0	0	0	0	0
17	5	5	5	5	5	5	5	5	5	2	2	2	0	0	0	0	0	0
18	5	5	5	5	5	5	5	5	5	3	3	3	0	0	0	0	0	0
19	5	5	5	5	5	5	5	5	5	4	4	4	0	0	0	0	0	0
20	5	5	5	5	5	5	5	5	5	5	5	5	0	0	0	0	0	0
21	5	5	5	5	5	5	5	5	5	5	5	5	1	1	1	1	1	1
22	5	5	5	5	5	5	5	5	5	5	5	5	2	2	2	2	2	2
23	5	5	5	5	5	5	5	5	5	5	5	5	3	3	3	3	3	3
24	5	5	5	5	5	5	5	5	5	5	5	5	4	4	4	4	4	4
25, 26, …	5	5	5	5	5	5	5	5	5	5	5	5	5	5	5	5	5	5

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表 2 最优值函数

Table 2. The optimal value function

u	β=0			β=0.2			β=1.2			不注资
u	V₁^*(u)	V₂^*(u)	V₃^*(u)	V₁^*(u)	V₂^*(u)	V₃^*(u)	V₁^*(u)	V₂^*(u)	V₃^*(u)	W₁^*(u)	W₂^*(u)	W₃^*(u)
0	34.650 1	35.061 4	35.033 0	29.566 9	29.938 8	29.912 0	8.898 5	9.126 3	9.102 4	10.566 7	10.673 5	10.667 2
1	35.650 1	36.061 4	36.033 0	30.766 9	31.138 8	31.112 0	11.098 5	11.326 3	11.302 4	11.740 8	11.859 4	11.852 5
2	36.650 1	37.061 4	37.033 0	31.966 9	32.338 8	32.312 0	13.298 5	13.526 3	13.502 4	13.045 4	13.177 2	13.169 4
3	37.650 1	38.061 4	38.033 0	33.166 9	33.538 8	33.512 0	15.498 5	15.726 3	15.702 4	14.494 9	14.641 3	14.632 7
4	38.650 1	39.061 4	39.033 0	34.366 9	34.738 8	34.712 0	17.698 5	17.926 3	17.902 4	16.105 4	16.268 1	16.258 5
5	39.650 1	40.061 4	40.033 0	35.566 9	35.938 8	35.912 0	19.898 5	20.126 3	20.102 4	20.153 1	20.357 0	20.344 9
6	40.563 5	40.981 1	40.953 3	36.611 9	36.990 9	36.964 8	21.621 3	21.861 2	21.838 3	21.514 1	21.731 7	21.718 8
7	41.471 9	41.896 0	41.868 7	37.645 3	38.031 5	38.006 1	23.301 5	23.553 2	23.531 4	22.928 8	23.160 6	23.146 9
8	42.374 9	42.805 5	42.778 7	38.665 7	39.059 2	39.034 4	24.934 2	25.197 8	25.176 8	24.392 3	24.638 7	24.624 2
9	43.271 8	43.708 9	43.682 6	39.671 9	40.072 7	40.048 5	26.514 1	26.789 7	26.769 4	25.897 9	26.159 4	26.144 0
10	44.161 9	44.605 7	44.579 8	40.662 1	41.070 4	41.046 6	28.035 4	28.322 8	28.303 1	27.919 5	28.201 3	28.184 9
11	45.026 6	45.477 8	45.452 1	41.602 2	42.018 6	41.995 0	29.391 2	29.691 0	29.671 3	29.159 1	29.453 2	29.436 1
12	45.888 8	46.347 3	46.321 8	42.532 9	42.957 1	42.933 7	30.704 5	31.016 4	30.996 6	30.402 4	30.708 8	30.691 0
13	46.748 7	47.214 4	47.189 1	43.453 8	43.885 9	43.862 6	31.974 3	32.298 0	32.278 1	31.643 1	31.961 7	31.943 4
14	47.606 4	48.079 3	48.054 2	44.365 1	44.804 8	44.781 8	33.199 6	33.534 7	33.514 7	32.874 3	33.204 9	33.186 1
15	48.462 3	48.942 3	48.917 3	45.266 9	45.714 1	45.691 2	34.380 0	34.726 1	34.705 8	34.191 7	34.535 2	34.515 8
16	49.341 0	49.827 5	49.802 9	46.164 5	46.618 4	46.596 0	35.493 8	35.850 4	35.829 9	35.272 3	35.626 3	35.606 4
17	50.210 9	50.704 0	50.679 8	47.055 5	47.516 2	47.494 1	36.577 4	36.944 3	36.923 6	36.341 4	36.705 5	36.685 3
18	51.071 1	51.570 8	51.546 9	47.939 0	48.406 6	48.384 8	37.631 9	38.008 7	37.987 7	37.395 8	37.770 0	37.749 4
19	51.920 3	52.426 9	52.403 2	48.814 2	49.289 0	49.267 4	38.658 4	39.044 8	39.023 6	38.432 8	38.816 7	38.795 8
20	52.757 5	53.271 1	53.247 6	49.680 3	50.162 3	50.140 8	39.658 4	40.054 0	40.032 6	39.471 9	39.865 5	39.844 3
21	53.553 2	54.074 4	54.050 7	50.524 1	51.014 3	50.992 6	40.629 3	41.033 7	41.012 2	40.435 0	40.837 4	40.816 1
22	54.344 3	54.873 2	54.849 3	51.359 2	51.857 5	51.835 6	41.582 6	41.995 6	41.974 0	41.387 3	41.798 3	41.776 8
23	55.130 8	55.667 3	55.643 1	52.185 7	52.692 0	52.669 8	42.519 7	42.941 1	42.919 4	42.328 0	42.747 4	42.725 8
24	55.912 3	56.456 4	56.432 0	53.003 5	53.517 7	53.495 4	43.442 4	43.871 9	43.850 2	43.256 8	43.684 4	43.662 8
25	56.688 7	57.240 3	57.215 7	53.812 9	54.335 0	54.312 3	44.352 3	44.789 6	44.767 9	44.178 5	44.614 0	44.592 5
26	57.464 8	58.023 8	57.999 1	54.614 9	55.144 7	55.121 9	45.255 6	45.699 7	45.678 4	45.082 5	45.524 8	45.503 6
27	58.231 4	58.797 9	58.772 9	55.408 8	55.946 3	55.923 3	46.152 9	46.603 9	46.582 9	45.980 5	46.429 6	46.408 8
28	58.988 2	59.562 1	59.536 9	56.194 2	56.739 4	56.716 1	47.043 5	47.501 4	47.480 7	46.871 8	47.328 0	47.307 4
29	59.735 0	60.316 3	60.290 7	56.970 8	57.523 7	57.500 0	47.926 5	48.391 7	48.371 1	47.755 7	48.219 2	48.198 7
30	60.471 5	61.060 1	61.034 2	57.738 3	58.298 8	58.274 8	48.801 3	49.273 9	49.253 4	48.631 4	49.102 3	49.081 9

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由表 1可知:β=0, 0.1, 0.4, 0.6, 1.2时, φ_i^*(u)的门槛值分别为10、10、12、16、20, 表明本例给出的最优分红策略φ_i^*(u)是Threshold策略^[2].由表 2知:在资金u相同的条件下, 该最优分红策略φ_i^*(u)所对应的最优值函数V_i^*(u)与交易费比例β有关, V_i^*(u)的值随着β的增大而减少; 再对比表 2中未注资的最优值函数W_i^*(u), 发现交易费比例β控制在一定范围内时V_i^*(u)明显大于W_i^*(u).这说明了本文的最优红利值算法的可行性, 也验证了本文的最优分红策略的有效性.

5. 小结

本文在复合二项对偶模型中讨论了带比例交易费再注资且分红贴现利率随机变化的最优分红问题, 运用压缩映射不动点原理证明了该最优分红问题的最优值函数是一个离散的HJB方程的唯一解, 得到了最优分红策略和最优值函数的优化算法.数值实例结果表明:当交易费比例控制在一定范围内时，相应的最优红利值大于未注资的最优红利值, 说明了文中最优分红策略的有效性.在今后的工作中, 可进一步研究最优分红策略的Threshold性质, 也可尝试在红利无限制的条件下讨论最优分红问题.

表 1 最优分红策略φ_i^*(u) (i=1, 2, 3, C=5)

Table 1 The optimal dividend strategy φ_i^*(u) (i=1, 2, 3, C=5)

u	β=0			β=0.2			β=0.4			β=0.6			β=1.2			不注资
u	i=1	i=2	i=3	i=1	i=2	i=3	i=1	i=2	i=3	i=1	i=2	i=3	i=1	i=2	i=3	i=1	i=2	i=3
0, 1, …, 10	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
11	1	1	1	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0
12	2	2	2	2	2	2	1	0	0	0	0	0	0	0	0	0	0	0
13	3	3	3	3	3	3	2	1	1	0	0	0	0	0	0	0	0	0
14	4	4	4	4	4	4	3	2	2	0	0	0	0	0	0	0	0	0
15	5	5	5	5	5	5	4	3	3	0	0	0	0	0	0	0	0	0
16	5	5	5	5	5	5	5	4	4	1	1	1	0	0	0	0	0	0
17	5	5	5	5	5	5	5	5	5	2	2	2	0	0	0	0	0	0
18	5	5	5	5	5	5	5	5	5	3	3	3	0	0	0	0	0	0
19	5	5	5	5	5	5	5	5	5	4	4	4	0	0	0	0	0	0
20	5	5	5	5	5	5	5	5	5	5	5	5	0	0	0	0	0	0
21	5	5	5	5	5	5	5	5	5	5	5	5	1	1	1	1	1	1
22	5	5	5	5	5	5	5	5	5	5	5	5	2	2	2	2	2	2
23	5	5	5	5	5	5	5	5	5	5	5	5	3	3	3	3	3	3
24	5	5	5	5	5	5	5	5	5	5	5	5	4	4	4	4	4	4
25, 26, …	5	5	5	5	5	5	5	5	5	5	5	5	5	5	5	5	5	5

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表 2 最优值函数

Table 2 The optimal value function

u	β=0			β=0.2			β=1.2			不注资
u	V₁^*(u)	V₂^*(u)	V₃^*(u)	V₁^*(u)	V₂^*(u)	V₃^*(u)	V₁^*(u)	V₂^*(u)	V₃^*(u)	W₁^*(u)	W₂^*(u)	W₃^*(u)
0	34.650 1	35.061 4	35.033 0	29.566 9	29.938 8	29.912 0	8.898 5	9.126 3	9.102 4	10.566 7	10.673 5	10.667 2
1	35.650 1	36.061 4	36.033 0	30.766 9	31.138 8	31.112 0	11.098 5	11.326 3	11.302 4	11.740 8	11.859 4	11.852 5
2	36.650 1	37.061 4	37.033 0	31.966 9	32.338 8	32.312 0	13.298 5	13.526 3	13.502 4	13.045 4	13.177 2	13.169 4
3	37.650 1	38.061 4	38.033 0	33.166 9	33.538 8	33.512 0	15.498 5	15.726 3	15.702 4	14.494 9	14.641 3	14.632 7
4	38.650 1	39.061 4	39.033 0	34.366 9	34.738 8	34.712 0	17.698 5	17.926 3	17.902 4	16.105 4	16.268 1	16.258 5
5	39.650 1	40.061 4	40.033 0	35.566 9	35.938 8	35.912 0	19.898 5	20.126 3	20.102 4	20.153 1	20.357 0	20.344 9
6	40.563 5	40.981 1	40.953 3	36.611 9	36.990 9	36.964 8	21.621 3	21.861 2	21.838 3	21.514 1	21.731 7	21.718 8
7	41.471 9	41.896 0	41.868 7	37.645 3	38.031 5	38.006 1	23.301 5	23.553 2	23.531 4	22.928 8	23.160 6	23.146 9
8	42.374 9	42.805 5	42.778 7	38.665 7	39.059 2	39.034 4	24.934 2	25.197 8	25.176 8	24.392 3	24.638 7	24.624 2
9	43.271 8	43.708 9	43.682 6	39.671 9	40.072 7	40.048 5	26.514 1	26.789 7	26.769 4	25.897 9	26.159 4	26.144 0
10	44.161 9	44.605 7	44.579 8	40.662 1	41.070 4	41.046 6	28.035 4	28.322 8	28.303 1	27.919 5	28.201 3	28.184 9
11	45.026 6	45.477 8	45.452 1	41.602 2	42.018 6	41.995 0	29.391 2	29.691 0	29.671 3	29.159 1	29.453 2	29.436 1
12	45.888 8	46.347 3	46.321 8	42.532 9	42.957 1	42.933 7	30.704 5	31.016 4	30.996 6	30.402 4	30.708 8	30.691 0
13	46.748 7	47.214 4	47.189 1	43.453 8	43.885 9	43.862 6	31.974 3	32.298 0	32.278 1	31.643 1	31.961 7	31.943 4
14	47.606 4	48.079 3	48.054 2	44.365 1	44.804 8	44.781 8	33.199 6	33.534 7	33.514 7	32.874 3	33.204 9	33.186 1
15	48.462 3	48.942 3	48.917 3	45.266 9	45.714 1	45.691 2	34.380 0	34.726 1	34.705 8	34.191 7	34.535 2	34.515 8
16	49.341 0	49.827 5	49.802 9	46.164 5	46.618 4	46.596 0	35.493 8	35.850 4	35.829 9	35.272 3	35.626 3	35.606 4
17	50.210 9	50.704 0	50.679 8	47.055 5	47.516 2	47.494 1	36.577 4	36.944 3	36.923 6	36.341 4	36.705 5	36.685 3
18	51.071 1	51.570 8	51.546 9	47.939 0	48.406 6	48.384 8	37.631 9	38.008 7	37.987 7	37.395 8	37.770 0	37.749 4
19	51.920 3	52.426 9	52.403 2	48.814 2	49.289 0	49.267 4	38.658 4	39.044 8	39.023 6	38.432 8	38.816 7	38.795 8
20	52.757 5	53.271 1	53.247 6	49.680 3	50.162 3	50.140 8	39.658 4	40.054 0	40.032 6	39.471 9	39.865 5	39.844 3
21	53.553 2	54.074 4	54.050 7	50.524 1	51.014 3	50.992 6	40.629 3	41.033 7	41.012 2	40.435 0	40.837 4	40.816 1
22	54.344 3	54.873 2	54.849 3	51.359 2	51.857 5	51.835 6	41.582 6	41.995 6	41.974 0	41.387 3	41.798 3	41.776 8
23	55.130 8	55.667 3	55.643 1	52.185 7	52.692 0	52.669 8	42.519 7	42.941 1	42.919 4	42.328 0	42.747 4	42.725 8
24	55.912 3	56.456 4	56.432 0	53.003 5	53.517 7	53.495 4	43.442 4	43.871 9	43.850 2	43.256 8	43.684 4	43.662 8
25	56.688 7	57.240 3	57.215 7	53.812 9	54.335 0	54.312 3	44.352 3	44.789 6	44.767 9	44.178 5	44.614 0	44.592 5
26	57.464 8	58.023 8	57.999 1	54.614 9	55.144 7	55.121 9	45.255 6	45.699 7	45.678 4	45.082 5	45.524 8	45.503 6
27	58.231 4	58.797 9	58.772 9	55.408 8	55.946 3	55.923 3	46.152 9	46.603 9	46.582 9	45.980 5	46.429 6	46.408 8
28	58.988 2	59.562 1	59.536 9	56.194 2	56.739 4	56.716 1	47.043 5	47.501 4	47.480 7	46.871 8	47.328 0	47.307 4
29	59.735 0	60.316 3	60.290 7	56.970 8	57.523 7	57.500 0	47.926 5	48.391 7	48.371 1	47.755 7	48.219 2	48.198 7
30	60.471 5	61.060 1	61.034 2	57.738 3	58.298 8	58.274 8	48.801 3	49.273 9	49.253 4	48.631 4	49.102 3	49.081 9

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1. 预备知识
2. 最优分红策略和最优值函数
3. 最优红利值的算法
4. 数值模拟
5. 小结

随机利率下带注资的对偶模型最优分红问题

通讯作者: 郑华, 副教授, Email:hzhengjlfdu@gmail.com

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