最优分红策略:正则与脉冲混合控制问题

Optimal dividend policy:A regular-impulse stochastic control problem

本文对扩散模型下的最优分红问题作了进一步分析.注意到,累积分红量是一个关于时间的右连左极过程,它的路径由连续和跳跃两部分组成.因此,本文在建模中同时加入了连续分红和脉冲分红两种形式,这就构成了一个正则和脉冲分红混合的最优控制问题.假设所有分红量存在一个比例成本,对于每次的脉冲分红量存在一个固定成本.此外,对于连续分红部分,假设存在一个有限的最大分红率.用漂移Brown运动描述公司的盈余过程,优化目标设定为最大化公司破产前分红现值的期望值,本文给出了值函数以及最优分红策略的解析表达式.结论表明,最优的分红策略为阀值(threshold)策略和脉冲策略的组合形式.

In this paper,we revisit the classic optimal dividend problem in a diffusion model. Note that,the aggregated dividend payments process is a cadlag process,whose path can be partitioned into two parts：Continuous part and discontinuous part. Therefore,we incorporate both the continuous dividend form and the impulse dividend form in our model,which leads to a regular-impulse combined stochastic control problem. We assume that there exist a maximum rate for continuous dividend payments and both proportional and fixed costs for impulse dividend payments. When the company＇s surplus is described by a drifted Brownian motion,and the objective is to maximize the expectation of total discounted dividend payments up to the time of ruin,we obtain the explicit closed-form expressions for the value function and the optimal dividend policy. The optimal dividend policy is shown to be a combinational form of the threshold policy and the impulse policy.