带有随机延迟的最优分红和资本注资问题

Optimal Dividend and Capital Injection Problem with Random Delay

本文考虑了带有注资延迟的最优分红问题,并且假设注资延迟服从指数分布.该问题的目标是找到最优的分红策略和注资策略使得分红效用以及注资效用达到最大.由于保险公司的盈余过程涉及到混合泊松过程,应用扩散近似原则,我们用一个随机微分方程来刻画盈余过程.当值函数足够光滑时,使用动态规划方法,我们得到相应的拟变分不等式.本文从三个不同的区域(即分红区域、连续区域和注资区域)来讨论值函数.通过边界条件,我们得到不同区域中值函数的表达式且给出了验证性定理.数值例子呈现了注资延迟在不同参数下的影响.

This article considers the optimal dividend policy with delayed capital injections,and assumes that the capital injection delay follows the exponential distribution.We aim to find the optimal dividend and capital injection strategies to maximize the utility of dividend and capital.Since surplus process of the insurance company involves a mixed Poisson process,we use a stochastic differential equation to characterize the surplus process by adopting diffusion approximation techniques,and then we obtain the value function under the utility criterion.When the value function is smooth,the quasi variational inequality is obtained by using the dynamic programming principle.In this paper,we consider the value function from three different regions(the dividend area,the continuous area and the capital injection area).Through the boundary conditions,we derive the expression of the value function in different regions and present the verification theorem.A numerical example is presented to illustrate the effects of the capital injection delay under different parameters.