﻿ 带约束复合泊松模型的最优分红策略 Optimal Dividend Strategy with Constrained Compound Poisson Model

Vol. 08  No. 03 ( 2019 ), Article ID: 29439 , 8 pages
10.12677/AAM.2019.83062

Optimal Dividend Strategy with Constrained Compound Poisson Model

Yanshuang Zheng, Guoxin Liu

School of Science, Hebei University of Technology, Tianjin

Received: Mar. 5th, 2019; accepted: Mar. 20th, 2019; published: Mar. 27th, 2019

ABSTRACT

In this paper, we study the optimal dividend problem for a compound Poisson risk model with constant interest rate under bounded dividend rate. We aim to maximize the expected cumulative discounted dividends before bankruptcy by using bounded dividend rate. And the relevant characteristics of the dividend strategy are given. Finally, based on the measure-valued generator theory, this paper deduces the associated measure-valued dynamic programming equation (DPE), and further analyzes the relationship between the measure-valued DPE and the quasi-variational inequality.

Keywords:Compound Poisson Model, Optimal Dividend, Bounded Dividend Rates, Measure-Valued DPE

1. 引言

2. 模型介绍

2.1. 基本模型

$\left(\Omega ,\mathcal{F},ℙ\right)$ 为全概率空间， $\Omega$ 为左连右极集合，其中所有随机变量均可在该空间上定义。对于任意 $t\ge 0,s\ge 0,\omega \in \Omega$，定义转移算子 ${\theta }_{t}:\Omega \to \Omega$${\left({\theta }_{t}\omega \right)}_{s}={\omega }_{s+t}$，则盈余过程 $X={\left\{{X}_{t}\right\}}_{t\ge 0}$ 可以表示为

${X}_{t}=x+ct-\underset{k=1}{\overset{{N}_{t}}{\sum }}{U}_{k}+i{\int }_{0}^{t}{X}_{s}\text{d}s,$ (1)

${X}_{t}^{L}=x+ct-\underset{k=1}{\overset{{N}_{t}}{\sum }}{U}_{k}+i{\int }_{0}^{t}{X}_{s}\text{d}s-{L}_{t}.$ (2)

${\tau }^{L}:=\mathrm{inf}\left\{t\ge 0:{X}_{t}^{L}<0\right\}$ 表示保险公司的破产时刻。给定分红速率上限 ${l}_{0}\ge 0$，则分红策略 $L={\left\{{L}_{t}\right\}}_{t<0}$ 是可行的，如果

· 过程L是非降适应 ${\left\{\mathcal{F}\right\}}_{t\ge 0}$ 的，且满足 ${L}_{0}=0$

· 对 $h\ge 0$，有 ${L}_{t+h}-{L}_{t}\le {l}_{0}h$

· 分红不会导致破产；

· SDE(2)有唯一强解。

${V}^{L}\left(x\right)=\mathbb{E}\left[{\int }_{0}^{{\tau }^{L}}{\text{e}}^{-\delta s}\text{d}{L}_{s}|{X}_{0}=x\right]={\mathbb{E}}_{x}\left[{\int }_{0}^{{\tau }^{L}}{\text{e}}^{-\delta s}\text{d}{L}_{s}\right],$ (3)

$V\left(x\right)=\mathrm{sup}\left\{{V}^{L}\left(x\right),L\in {\Pi }_{x}\right\},x\ge 0.$ (4)

2.2. 值函数

i) $\forall y>x\ge 0,\exists V\left(x\right)\le V\left(y\right)$

ii) $\forall x\ge 0,\exists V\left(x\right)\le {l}_{0}/\delta$

iii) $V\left(x\right)$ 是局部利普西茨连续的。

$V\left(x\right)\ge {V}^{\stackrel{˜}{L}}\left(x\right)\ge {\text{e}}^{-\left(\lambda +\delta \right){t}_{0}}{V}^{L}\left(y\right)\ge {\text{e}}^{-\left(\lambda +\delta \right){t}_{0}}\left(V\left(y\right)-ϵ\right).$

3. DPP&DPE

3.1. 动态规划原理

$V\left(x\right)=\underset{l\in \mathbb{M}}{\mathrm{sup}}{\mathbb{E}}_{x}\left[{\int }_{0}^{t\wedge {\tau }_{1}}{\text{e}}^{-\delta s}l\left({X}_{s}^{l}\right)\text{d}s+{\text{e}}^{-\delta \left(t\wedge {\tau }_{1}\right)}V\left({X}_{t\wedge {\tau }_{1}}^{l}\right)\right].$

$V\left(x\right)={V}^{{L}^{*}}\left(x\right)={\mathbb{E}}_{x}\left[{\int }_{0}^{t\wedge {\tau }_{1}}{\text{e}}^{-\delta s}{l}^{*}\left({X}_{s}^{{l}^{*}}\right)\text{d}s\right]+{\mathbb{E}}_{x}\left[{\int }_{t\wedge {\tau }_{1}}^{{\tau }^{L{}^{*}}}{\text{e}}^{-\delta s}{l}^{*}\left({X}_{s}^{{l}^{*}}\right)\text{d}s\right]$

$\begin{array}{l}{\mathbb{E}}_{x}\left[{\int }_{t\wedge {\tau }_{1}}^{{\tau }^{L{}^{*}}}{\text{e}}^{-\delta s}{l}^{*}\left({X}_{s}^{{l}^{*}}\right)\text{d}s\right]\\ ={\mathbb{E}}_{x}\left[\mathbb{E}\left({\int }_{t\wedge {\tau }_{1}}^{{\tau }^{L{}^{*}}}{\text{e}}^{-\delta s}{l}^{*}\left({X}_{s}^{{l}^{*}}\right)\text{d}s|{\mathcal{F}}_{t\wedge {\tau }_{1}}\right)\right]\\ ={\mathbb{E}}_{x}\left[\mathbb{E}\left({\int }_{t\wedge {\tau }_{1}}^{{\tau }^{L{}^{*}}}{\text{e}}^{-\delta s}{l}^{*}\left({X}_{s}^{{l}^{*}}\right)\text{d}s|{X}_{t\wedge {\tau }_{1}}^{{l}^{*}}\right)\right]\\ ={\mathbb{E}}_{x}\left[{\text{e}}^{-\delta \left(t\wedge {\tau }_{1}\right)}{\mathbb{E}}_{{X}_{t\wedge {\tau }_{1}}^{{l}^{*}}}\left({\int }_{0}^{{\tau }^{L{}^{*}}}{\text{e}}^{-\delta s}{l}^{*}\left({X}_{s}^{{l}^{*}}\right)\text{d}s\right)\right]\\ ={\mathbb{E}}_{x}\left[{\text{e}}^{-\delta \left(t\wedge {\tau }_{1}\right)}V\left({X}_{t\wedge {\tau }_{1}}^{{l}^{*}}\right)\right].\end{array}$

$V\left(x\right)={\mathbb{E}}_{x}\left[{\int }_{0}^{t\wedge {\tau }_{1}}{\text{e}}^{-\delta s}{l}^{*}\left({X}_{s}^{{l}^{*}}\right)\text{d}s+{\text{e}}^{-\delta \left(t\wedge {\tau }_{1}\right)}V\left({X}_{t\wedge {\tau }_{1}}^{{l}^{*}}\right)\right].$ (5)

${\stackrel{^}{L}}_{s}=\left\{\begin{array}{l}{\int }_{0}^{s}l\left({X}_{s}^{\stackrel{^}{L}}\right)\text{d}s,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\le s\le t\wedge {\tau }_{1};\\ {\stackrel{^}{L}}_{t\wedge {\tau }_{1}}+{\int }_{t\wedge {\tau }_{1}}^{{\tau }^{L{}^{*}}}{l}^{*}\left({X}_{s}^{\stackrel{^}{L}}\right)\text{d}s,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\wedge {\tau }_{1}

$V\left(x\right)\ge {V}^{\stackrel{^}{L}}\left(x\right)={\mathbb{E}}_{x}\left[{\int }_{0}^{t\wedge {\tau }_{1}}{\text{e}}^{-\delta s}l\left({X}_{s}^{l}\right)\text{d}s+{\int }_{t\wedge {\tau }_{1}}^{{\tau }^{\stackrel{^}{L}}}{\text{e}}^{-\delta s}{l}^{*}\left({X}_{s}^{{l}^{*}}\right)\text{d}s\right].$

${X}_{t\wedge {\tau }_{1}}^{\stackrel{^}{L}}={X}_{t\wedge {\tau }_{1}}^{l}$ 。根据推导(5)的方法，同样我们可以得到

$V\left(x\right)\ge {\mathbb{E}}_{x}\left[{\int }_{0}^{t\wedge {\tau }_{1}}{\text{e}}^{-\delta s}l\left({X}_{s}^{l}\right)\text{d}s+{\text{e}}^{-\delta \left(t\wedge {\tau }_{1}\right)}V\left({X}_{t\wedge {\tau }_{1}}^{l}\right)\right].$

3.2. 动态规划方程

$0=\underset{l\in \mathbb{M}}{\mathrm{sup}}\left\{{\mathcal{L}}^{l}V\left(x,t\right)+{\int }_{0}^{t}l\left({\varphi }_{x}^{l}\left(s\right)\right)\text{d}s-\delta {\int }_{0}^{t}V\left({\varphi }_{x}^{l}\left(s\right)\right)\text{d}s\right\},$ (6)

$\begin{array}{c}V\left(x\right)\ge {\mathbb{E}}_{x}\left[{\int }_{0}^{t\wedge {\tau }_{1}}{\text{e}}^{-\delta s}l\left({X}_{s}^{l}\right)\text{d}s+{\text{e}}^{-\delta \left(t\wedge {\tau }_{1}\right)}V\left({X}_{t\wedge {\tau }_{1}}^{l}\right)\right]\\ ={\text{e}}^{-\lambda t}{\int }_{0}^{t}{\text{e}}^{-\delta s}l\left({X}_{s}^{l}\right)\text{d}s+{\int }_{0}^{t}\lambda {\text{e}}^{-\lambda s}{\int }_{0}^{s}{\text{e}}^{-\delta u}l\left({X}_{u}^{l}\right)\text{d}u\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\text{e}}^{-\left(\lambda +\delta \right)t}V\left({\varphi }_{x}^{l}\left(s\right)\right)+{\int }_{0}^{t}\lambda {\text{e}}^{-\left(\lambda +\delta \right)s}\mathcal{Q}V\left({\varphi }_{x}^{l}\left(s\right)\right)\text{d}s.\end{array}$

$\begin{array}{l}{\text{e}}^{-\lambda t}{\int }_{0}^{t}{\text{e}}^{-\delta s}l\left({X}_{s}^{l}\right)\text{d}s\\ ={\int }_{0}^{t}{\text{e}}^{-\left(\lambda +\delta \right)s}l\left({X}_{s}^{l}\right)\text{d}s-{\int }_{0}^{t}\lambda {\text{e}}^{-\lambda s}{\int }_{0}^{s}{\text{e}}^{-\delta u}l\left({X}_{u}^{l}\right)\text{d}u\text{d}s\end{array}$

$\begin{array}{l}{\text{e}}^{-\left(\lambda +\delta \right)t}V\left({\varphi }_{x}^{l}\left(t\right)\right)-V\left(x\right)\\ ={\int }_{0}^{t}{\text{e}}^{-\left(\lambda +\delta \right)s}\text{d}V\left({\varphi }_{x}^{l}\left(s\right)\right)-\left(\lambda +\delta \right){\int }_{0}^{t}{\text{e}}^{-\left(\lambda +\delta \right)s}V\left({\varphi }_{x}^{l}\left(s\right)\right)\text{d}s.\end{array}$

$\begin{array}{c}0\ge {\int }_{0}^{t}{\text{e}}^{-\left(\lambda +\delta \right)s}\text{d}V\left({\varphi }_{x}^{l}\left(s\right)\right)+{\int }_{0}^{t}{\text{e}}^{-\left(\lambda +\delta \right)s}l\left({\varphi }_{x}^{l}\left(s\right)\right)\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\left(\lambda +\delta \right){\int }_{0}^{t}{\text{e}}^{-\left(\lambda +\delta \right)s}V\left({\varphi }_{x}^{l}\left(s\right)\right)\text{d}s+{\int }_{0}^{t}\lambda {\text{e}}^{-\left(\lambda +\delta \right)s}\mathcal{Q}V\left({\varphi }_{x}^{l}\left(s\right)\right)\text{d}s\\ ={\int }_{0}^{t}{\text{e}}^{-\left(\lambda +\delta \right)s}{\mathcal{H}}^{l}V\left(x,\text{d}s\right),\end{array}$ (7)

$\begin{array}{l}{\int }_{0}^{t+h}{\text{e}}^{-\left(\lambda +\delta \right)s}{\mathcal{H}}^{l}V\left(x,\text{d}s\right)\\ ={\int }_{0}^{t}{\text{e}}^{-\left(\lambda +\delta \right)s}{\mathcal{H}}^{l}V\left(x,\text{d}s\right)+{\int }_{t}^{t+h}{\text{e}}^{-\left(\lambda +\delta \right)s}{\mathcal{H}}^{l}V\left(x,\text{d}s\right)\\ ={\int }_{0}^{t}{\text{e}}^{-\left(\lambda +\delta \right)s}{\mathcal{H}}^{l}V\left(x,\text{d}s\right)+{\text{e}}^{-\left(\lambda +\delta \right)t}{\int }_{0}^{h}{\text{e}}^{-\left(\lambda +\delta \right)s}{\mathcal{H}}^{l}V\left({\varphi }_{x}^{l}\left(t\right),\text{d}s\right)\text{\hspace{0.17em}}.\end{array}$

$0\ge {\mathcal{L}}^{l}V\left(x,t\right)+{\int }_{0}^{t}l\left({\varphi }_{x}^{l}\left(s\right)\right)\text{d}s-\delta {\int }_{0}^{t}V\left({\varphi }_{x}^{l}\left(s\right)\right)\text{d}s.$ (8)

$0={\mathcal{L}}^{{l}^{*}}V\left(x,t\right)+{\int }_{0}^{t}{l}^{*}\left({\varphi }_{x}^{{l}^{*}}\left(s\right)\right)\text{d}s-\delta {\int }_{0}^{t}V\left({\varphi }_{x}^{{l}^{*}}\left(s\right)\right)\text{d}s.$ (9)

$0=\underset{l\in \mathbb{M}}{\mathrm{sup}}\left\{{\mathcal{L}}^{l}V\left(x,t\right)+{\int }_{0}^{t}l\left({\varphi }_{x}^{l}\left(s\right)\right)\text{d}s-\delta {\int }_{0}^{t}V\left({\varphi }_{x}^{l}\left(s\right)\right)\text{d}s\right\}.$ (10)

${\mathbb{U}}_{x}$ 表示函数 $\alpha :{ℝ}_{+}↦\left[0,{l}_{0}\right]$ 的集合且满足 ${\varphi }_{x}^{\alpha }\left(t\right)=x+{\int }_{0}^{t}\left[c+i{\varphi }_{x}^{\alpha }\left(s\right)\right]\text{d}s$ 。现在对于任意固定的 $x\ge 0$，根据(6)有

$0=\underset{\alpha \in {\mathbb{U}}_{x}}{\mathrm{sup}}\left\{{\mathcal{L}}^{l}V\left(x,t\right)+{\int }_{0}^{t}\alpha \left(s\right)\text{d}s-\delta {\int }_{0}^{t}V\left({\varphi }_{x}^{\alpha }\left(s\right)\right)\text{d}s\right\}.$ (11)

4. 测度值DPE与QVI的关系

$\begin{array}{l}0=\underset{l\in \mathbb{M}}{\mathrm{sup}}\left\{\left(1-{V}^{\prime }\left({\varphi }_{x}^{l}\left(t\right)\right)\right)l\left({\varphi }_{x}^{l}\left(t\right)\right)+{V}^{\prime }\left({\varphi }_{x}^{l}\left(t\right)\right)\left(c+i{\varphi }_{x}^{l}\left(t\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-\left(\lambda +\delta \right)V\left({\varphi }_{x}^{l}\left(t\right)\right)+\lambda \mathcal{Q}V\left({\varphi }_{x}^{l}\left(t\right)\right)\right\}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}a.e.\text{\hspace{0.17em}}\text{w}.\text{r}.\text{t}.\text{\hspace{0.17em}}t.\end{array}$ (12)

${V}^{\prime }\left(x\right)=\left\{\begin{array}{l}{V}^{\prime }\left(x\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}导数存在;\\ 1，\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}其他.\end{array}$

$\begin{array}{c}V\left({\varphi }_{x}^{l}\left(t\right)\right)-V\left(x\right)={\int }_{0}^{{\varphi }_{x}^{l}\left(t\right)}{V}^{\prime }\left(u\right)\text{d}u={\int }_{0}^{t}{V}^{\prime }\left({\varphi }_{x}^{l}\left(s\right)\right)\text{d}{\varphi }_{x}^{l}\left(s\right)\\ ={\int }_{0}^{t}{V}^{\prime }\left({\varphi }_{x}^{l}\left(s\right)\right)\left(c+i{\varphi }_{x}^{l}\left(s\right)-l\left({\varphi }_{x}^{l}\left(s\right)\right)\right)\text{d}s.\end{array}$

$\begin{array}{c}0=\underset{l\in \mathbb{M}}{\mathrm{sup}}\left\{{\int }_{0}^{t}\left[\left(1-{V}^{\prime }\left({\varphi }_{x}^{l}\left(s\right)\right)\right)l\left({\varphi }_{x}^{l}\left(s\right)\right)+{V}^{\prime }\left({\varphi }_{x}^{l}\left(s\right)\right)\left(c+i{\varphi }_{x}^{l}\left(s\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\left(\lambda +\delta \right)V\left({\varphi }_{x}^{l}\left(s\right)\right)+\lambda \mathcal{Q}V\left({\varphi }_{x}^{l}\left(s\right)\right)\right]\text{d}s\right\}\\ =\underset{l\in \mathbb{M}}{\mathrm{sup}}\left\{{\int }_{0}^{t}{\mathcal{G}}^{l}V\left({\varphi }_{x}^{l}\left(s\right)\right)\text{d}s\right\},\end{array}$ (13)

${\int }_{0}^{t}{\mathcal{G}}^{l}V\left({\varphi }_{x}^{l}\left(s\right)\right)\text{d}s\le 0.$ (14)

$\begin{array}{c}{\int }_{0}^{t+h}{\mathcal{G}}^{l}V\left({\varphi }_{x}^{l}\left(s\right)\right)\text{d}s={\int }_{0}^{h}{\mathcal{G}}^{l}V\left({\varphi }_{x}^{l}\left(s\right)\right)\text{d}s+{\int }_{0}^{t+h}{\mathcal{G}}^{l}V\left({\varphi }_{x}^{l}\left(s\right)\right)\text{d}s\\ ={\int }_{0}^{h}{\mathcal{G}}^{l}V\left({\varphi }_{x}^{l}\left(s\right)\right)\text{d}s+{\int }_{0}^{t}{\mathcal{G}}^{l}V\left({\varphi }_{x}^{l}\left(h+s\right)\right)\text{d}s\\ ={\int }_{0}^{h}{\mathcal{G}}^{l}V\left({\varphi }_{x}^{l}\left(s\right)\right)\text{d}s+{\int }_{0}^{t}{\mathcal{G}}^{l}V\left({\varphi }_{{\varphi }_{x}^{l}\left(h\right)}^{l}\left(s\right)\right)\text{d}s.\end{array}$

$0=\underset{l\in \mathbb{M}}{\mathrm{sup}}\left\{{\int }_{0}^{t}{\mathcal{G}}^{l}V\left({\varphi }_{x}^{l}\left(s\right)\right)\text{d}s\right\}\le {\int }_{0}^{t}\underset{l\in \mathbb{M}}{\mathrm{sup}}{\mathcal{G}}^{l}V\left({\varphi }_{x}^{l}\left(s\right)\right)\text{d}s\le 0,$

$\Lambda V\left(x\right):=c+ix-\left(\lambda +\delta \right)V\left(x\right)+\lambda \mathcal{Q}V\left(x\right);$

$\mathcal{A}V\left(x\right):=\left(1-{V}^{\prime }\left(x\right)\right){l}_{0}+\left(c+ix\right){V}^{\prime }\left(x\right)-\left(\lambda +\delta \right)V\left(x\right)+\lambda \mathcal{Q}V\left(x\right);$

$\mathcal{B}V\left(x\right):=\left(c+ix\right){V}^{\prime }\left(x\right)-\left(\lambda +\delta \right)V\left(x\right)+\lambda \mathcal{Q}V\left(x\right).$

${\mathbb{D}}_{1}=\left\{x\in {ℝ}_{+}:c+ix<{l}_{0}\right\}$

${\mathbb{D}}_{2}=\left\{x\in {ℝ}_{+}:c+ix={l}_{0}\right\}$

${\mathbb{D}}_{3}=\left\{x\in {ℝ}_{+}:c+ix>{l}_{0}\right\}.$

$\left\{\begin{array}{l}\text{Λ}V\left(x\right)\le 0;\\ \mathcal{A}V\left(x\right)\le 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{a}.\text{e}.;\\ \mathcal{B}V\left(x\right)\le 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{a}.\text{e}.;\\ \left(\text{Λ}V\left(x\right)\right)\left(\mathcal{A}V\left(x\right)\right)\left(\mathcal{B}V\left(x\right)\right)=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{a}.\text{e}.\end{array}$

$x\in {\mathbb{D}}_{3}$，则值函数V满足QVI(I

$\left\{\begin{array}{l}\mathcal{A}V\left(x\right)\le 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{a}.\text{e}.;\\ \mathcal{B}V\left(x\right)\le 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{a}.\text{e}.;\\ \left(\mathcal{A}V\left(x\right)\right)\left(\mathcal{B}V\left(x\right)\right)=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{a}.\text{e}.\end{array}$

$0=\underset{a\in \left[0,{l}_{0}\right]}{\mathrm{sup}}\left\{a\left(1-{V}^{\prime }\left(x\right)\right)+\left(c+ix\right){V}^{\prime }\left(x\right)-\left(\lambda +\delta \right)V\left(x\right)+\lambda \mathcal{Q}V\left(x\right)\right\}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{a}.\text{e}.$ (15)

$\Lambda V\left(x\right)

$\mathcal{B}V\left(x\right)=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{a}.\text{e}.$

$\Lambda V\left(x\right)=\mathcal{A}V\left(x\right)=\mathcal{B}V\left(x\right).$

$\Lambda V\left(x\right)=\mathcal{A}V\left(x\right)=\mathcal{B}V\left(x\right)=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{a}.\text{e}.$

$\Lambda V\left(x\right)\le \mathcal{A}V\left(x\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathcal{B}V\left(x\right)

$\mathcal{A}V\left(x\right)=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{a}.\text{e}.$

$\left\{x\in {\mathbb{D}}_{3}:{V}^{\prime }\left(x\right)>1\right\}$$\mathcal{B}V\left(x\right)=0,\text{\hspace{0.17em}}\text{a}.\text{e}.$，且 $\mathcal{A}V\left(x\right)

$\left\{x\in {\mathbb{D}}_{3}:{V}^{\prime }\left(x\right)=1\right\}$$\mathcal{A}V\left(x\right)=\mathcal{B}V\left(x\right)=0,\text{\hspace{0.17em}}\text{a}.\text{e}.$

$\left\{x\in {\mathbb{D}}_{3}:{V}^{\prime }\left(x\right)<1\right\}$$\mathcal{A}V\left(x\right)=0,\text{\hspace{0.17em}}\text{a}.\text{e}.$，且 $\mathcal{B}V\left(x\right)<\mathcal{A}V\left(x\right)$

Optimal Dividend Strategy with Constrained Compound Poisson Model[J]. 应用数学进展, 2019, 08(03): 561-568. https://doi.org/10.12677/AAM.2019.83062

1. 1. De Finetti, B. (1957) Su un’Impostazione Alternativa Della Teoria Collettiva del Rischio. Transactions of the XVth In-ternational Congress of Actuaries, 2, 433-443.

2. 2. Gerber, H.U. and Shiu, E.S. (2006) On Optimal Dividend: From Reflection to Refraction. Journal of Computational and Applied Mathematics, 186, 4-22. https://doi.org/10.1016/j.cam.2005.03.062

3. 3. Azcue, P. and Muler, N. (2012) Optimal Dividend Policies for Compound Poisson Processes: The Case of Bounded Dividend Rates. Insurance: Mathematics and Economics, 51, 26-42. https://doi.org/10.1016/j.insmatheco.2012.02.011

4. 4. Gerber, H.U. and Shiu, E.S.W. (2006) On Optimal Dividend Strategies in the Compound Poisson Model. North American Actuarial Journal, 10, 76-93. https://doi.org/10.1080/10920277.2006.10596249

5. 5. Schmidli, H. (2008) Stochastic Control in Insurance. Springer, Berlin.

6. 6. Albrecher, H. and Thonhauser, S. (2008) Optimal Dividend Strategies for a Risk Process under Force of Interest. Insurance: Mathematics and Economics, 43, 134-149. https://doi.org/10.1016/j.insmatheco.2008.03.012

7. 7. Cai, J., Gerber, H.U. and Yang, H. (2006) Optimal Divi-dends in an Ornstein-Uhlenbeck Type Model with Credit and Debit Interest. North American Actuarial Journal, 10, 94-108. https://doi.org/10.1080/10920277.2006.10596250

8. 8. Fang, Y. and Wu, R. (2007) Optimal Dividend Strategy in the Compound Poisson Model with Constant Interest. Stochastic Models, 23, 149-166. https://doi.org/10.1080/15326340601142271

9. 9. Zhu, J. (2015) Dividend Optimization for General Diﬀusions with Restricted Dividend Payment Rates. Scandinavian Actuarial Journal, 2015, 592-615. https://doi.org/10.1080/03461238.2013.872174

10. 10. Liu, Z., Jiao, Y. and Liu, G. (2017) Measure-Valued Gener-ators of General Piecewise Deterministic Markov Processes. arXiv:1704.00938