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

作　　者：汪浩 程孝强 宫小洁 WANG Hao;CHENG Xiaoqiang;GONG Xiaojie(School of Mathematics and Statistics,Anhui Normal University,Wuhu,241002,China;School of Mathematics and Finance,Anhui Polytechnic University,Wuhu,241000,China;School of Statistics,East China Normal University,Shanghai,200062,China)

机构地区：[1]安徽师范大学数学与统计学院,芜湖241002 [2]安徽工程大学数理与金融学院,芜湖241000 [3]华东师范大学统计学院,上海200062

出　　处：《应用概率统计》2022年第2期267-284,共18页Chinese Journal of Applied Probability and Statistics

基　　金：安徽省高等学校自然科学研究项目(批准号:KJ2021A0104,SK2021A0284,KJ2021A0107)资助.

摘　　要：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.

关 键 词：最优分红策略 最优注资策略 动态规划原理 拟变分不等式 验证性定理 

分 类 号：O211.9[理学—概率论与数理统计]