生存概率控制下的周期性红利优化问题

Periodic Dividend Optimization Problem Under the Control of Survival Probability

文章主要在带有利息收益的离散时间盈余模型中,在生存概率和有界红利率的约束条件下,讨论周期性红利优化问题：最大化破产前累积的周期性支付的红利现值的期望,并获得最优红利策略.假设在每个单位时间内收到的保费是正实值随机变量,且保费序列构成一个马尔科夫链.此外,我们还假设任意单位时间内索赔发生的概率和相应单位时间内收到的保费相关.首先,给定生存概率的约束条件,得到了红利支付的约束门槛.然后,通过变换值函数和运用不动点原理,得到了最优红利策略的一些性质和算法.最后通过数值实例解释该算法,并讨论生存概率对最优红利策略的影响.数值结果显示,最优红利策略是一个条件多门槛策略.这为现代企业（尤其是保险和金融公司）的决策者在兼顾和平衡公司健康发展与股东利益而进行红利决策和定量分析时提供了理论依据.

This paper considers the periodic dividend optimization problem under the constraints of survival probability and bounded dividend rates in a discrete-time surplus model with interest. The optimization objective is to maximize the cumulative expected present value of periodic dividends until ruin, and obtain the optimal policy. We assume that the premium received per unit time is a positive real-valued random variable, and the sequence of premiums is a Markov chain. In any time interval the probability of a claim occurrence is related to the premium received in the corresponding unit time. First, we obtain the constraint thresholds of dividend payments from a given constraint condition of survival probability. Then, we provide some properties and an algorithm for the optimal dividend strategy by transforming the value function and applying the fixed point theorem. Finally, a numerical example is provided to illustrate the performance of the transformation method and show the effect of survival probability on optimal strategy. The numerical results show that the optimal strategy is a conditional multi-threshold strategy. The algorithm and results provide a theoretical basis for the decision makers of modern enterprises（especially insurance and finance companies） to make dividend decisions and quantitative analysis, who want to take into account both the healthy development of the company and the interests of shareholders.