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复合二项对偶模型的最优分红问题

2015-03-26邓丽激扬

经济数学 2014年4期

邓丽 激扬

摘要研究复合二项对偶模型的最优分红问题, 通过分析HJB方程得到了最优分红策略和相应的最优值函数之间的关系以及最优值函数的简单计算方法. 通过讨论最优红利策略的一些性质得到了最优值函数的可无限逼近的上界和下界.

关键词对偶模型;HJB方程;压缩映射;最优分红策略

中图分类号O211.6 文献标识码A

AbstractThis paper discussed the problem of optimal dividendpayment in compound binomial dual model. The relationship between the optimal dividend strategy and the corresponding optimal value function was found by analysing the HJB equation, and a simple algorithm was obtained for calculating the optimal value function. From the properties of the optimal dividend strategy, an upper bound and a lower bound of the optimal value function were derived.

Key wordsdual model; HJB equation; contraction mapping; optimal dividend strategy

1引言

分红问题的提出可以追溯到De Finetti1在纽约第15届国际精算师大会上发表的一篇文章,他认为在风险模型中考虑分红更切实际. 目前研究得最多的分红策略有:Barrier策略2-4和Threshold策略5-9. 随着金融管理、公司业务和保险业务的发展,经典风险模型的对偶模型越来越受到重视10-14, 讨论相对较多的是连续时间经典风险模型的最优分红问题,例如:Avanzi等10运用拉普拉斯变换方法讨论了复合Poisson对偶模型的最优红利Barrier的确定方法;Gerber等11讨论了复合Poisson对偶模型的最优红利Barrier的一些近似方法. 然而离散时间的最优分红问题显然还没有得到足够的重视,尽管De Finetti11最开始讨论红利问题就是在一个离散模型中. 对偶模型可描述为:

本文研究复合二项对偶模型的最优分红问题,发现最优值函数满足一个离散的哈密顿-雅可比 -贝尔曼(HJB)方程,并运用压缩映射原理证明最优值函数是这个方程的唯一解,从而得到了最优分红策略的计算方法. 通过讨论最优红利策略的一些性质本文构造出了最优值函数的可无限逼近的一个上界和一个下界,以便能运用递归算法在Matlab中进行数值计算.

2基本模型假设

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