牛祥秋[*]
具有相依利率的离散时间风险模型破产概率的上界
牛祥秋[*]
(辽宁师范大学 数学学院,辽宁 大连 116029)
研究具有相依利息率的离散时间风险模型的破产概率,在模型中假定利率为一阶自回归结构,并且考虑风险投资.利用递归更新方法和鞅方法分别给出了破产概率的上界估计,并且讨论了相应的最小上界问题.
一阶自回归;破产概率;最优化投资;上界
近年来,越来越多的学者运用随机控制理论研究保险风险管理问题[1-3],但关于离散时间风险模型下的最优控制问题的文献相对较少.文献[4]研究了一类具有马儿可夫链利率和随机投资回报的离散时间风险过程,分析了破产概率的最小上界问题.文献[5]研究了2种类型破产概率的广义Lundberg不等式.本文研究具有一阶自回归利率结构的离散时间风险过程,并利用文献[4]中的证明方法,获得破产概率的上界估计,并讨论保险公司的最优投资策略问题.
1 模型描述
本文考虑离散时间风险模型
显然
2 运用递归更新方法确定破产概率的上界
Browne发现,在布朗运动模型下,最优的投资策略是常数策略.Gaier给出的破产概率最小上界的最优投资策略是一个渐进最优的常数策略.受这些结果的启发,文中只关注常数投资策略,即是一个常数,记为.为了简便,省略和的上角标.
使用归纳法,假设
由式(7)和式(8)可知
3 利用鞅方法确定破产概率的上界
由文献[5]可知,投资的决策函数是一个仅与变量有关的函数,独立于当前盈余水平.由文献[6]可知,最优控制策略是平稳的,因此允许存在这样一个集合,使得
引理1[4]812假设是一个非负测度函数,是关于代数的一个测度,则.并且,如果独立于,那么.
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Upper bound of ruin probability under discrete risk model with dependent rates of interest
NIU Xiang-qiu
(School of Mathematics,Liaoning Normal University,Dalian 116029,China)
Researched on the ruin probability of a discrete time risk model with dependent rates of interest.In this model,the interest rate was assumed to be a structure of first-order autoregression and risk investment was considered.The upper bounds were derived by renewal recursive technique and martingale method respectively,the corresponding minimum upper bound for ruin probability was also discussed.
first-order autoregression;ruin probability;optimal investment strategy;upper bound
F224
A doi:10.3969/j.issn.1007-9831.2016.01.007
2015-11-20
牛祥秋(1989-),男,山东菏泽人,在读硕士研究生,从事保险精算研究.E-mail:657049682@qq.com