考虑随机观察时间的马尔可夫到达风险模型的分红问题

The risk model with dividend strategy has become one of the hot topics in risk theory. The project consider dividend problems in risk model where claims arrive according to a Markovian arrive process, the main contents include: (1)We consider a MAP risk model with exponentially distributed observation time under barrier strategy, and we will derive and solve the integro-differential equations for the Gerber-Shiu expected discounted penalty function and the total dividend payments, then we present some asymptotic formulas for the Gerber-Shiu expected discounted penalty function and the total dividend payments until ruin when the claim size is subexponential distribution of S; (2)We consider a MAP risk model with Erlang(n) distributed observation time under barrier strategy, and we will derive and solve the integro-differential equations for the Gerber-Shiu expected discounted penalty function and the total dividend payments, then we present some asymptotic formulas for the Gerber-Shiu expected discounted penalty function and the total dividend payments when the claims size is subexponential distribution of S; (3)We consider dividend problems on the perturbed MAP risk model under exponentially distributed observation time or Erlang(n) distributed separately, and then we will derive and solve the integro-differential equations for the Gerber-Shiu expected discounted penalty function and the total dividend payments, and then present some asymptotic formulas for the Gerber-Shiu expected discounted penalty function and the total dividend payments when the claim size is subexponential distribution of S.

近年来分红策略下的风险模型成为风险理论的研究热点之一。本项目拟研究带随机观察时间的马尔可夫到达风险模型的分红问题，主要内容包括：(1)考虑观察时间为指数分布的障碍分红策略下的马尔可夫到达风险模型，讨论Gerber-Shiu折现罚函数和累积分红折现均值满足的积分微分方程及其解，并研究索赔额属于次指数分布族S时相应的渐进公式；(2)考虑观察时间为Erlang(n)分布的障碍分红策略下的马尔可夫到达风险模型，讨论Gerber-Shiu折现罚函数和累积分红折现均值满足的积分微分方程及其解，并研究索赔额属于次指数分布族S时相应的渐进公式；(3)分别考虑观察时间为指数分布和Erlang(n)分布的障碍分红策略下带扰动的马尔可夫到达风险模型，讨论Gerber-Shiu折现罚函数和累积分红折现均值满足的积分微分方程及其解，并研究索赔额属于次指数分布族S时相应的渐进公式。

(1)考虑观察时间为指数分布的障碍分红策略下的两离散相依险种的马尔可夫风险模型的分红问题，其中险种的相依性是假定一个险种的主索赔以一定的概率引起另外一险种的副索赔, 且副索赔可能延迟发生, 推导了到破产前一时刻为止累积分红折现均值满足的差分方程, 并得到了索赔额分布是有理函数时累积分红折现均值和破产概率的的具体表达式, 并结合实际例子进行了数值模拟.(2) 考虑随机观察时间的常数分红策略和有流动盈余准备金的马尔可夫风险模型的分红问题，在绝对破产概率下得到了破产前累积分红折现均值以及破产概率的具体表达式，并对指数索赔下破产前累积分红折现均值进行了数值模拟，讨论了模型中相关参数对累积分红折现均值的影响。