分形布朗运动下最优投保和消费策略

Optimal strategies on portfolio and consumption with insurance under fractional Brownian motion

基于Merton的最优消费和投资组合模型,通过假设风险资产的价格变化服从几何分形布朗运动,探讨了一类具有人寿保险的最优投资消费问题.首先根据投资者在整个生命周期的消费和投保效用期望值最大的原则,利用贝尔曼动态规划原理,建立了最优投保和消费策略模型.然后在给定消费和遗赠评价效用函数的情况下,给出了最优投保和消费的闭式解,并获得了最优投资组合受模型参数变化影响的一些重要性质.最后,通过数值例子讨论了时间间隔、赫斯特指数变化时最优投保和最大期望效用的变化趋势.

Based on Merton＇ s optimal consumption and investment model, this paper researches a class of optimal portfolio and consumption problem that combines life insurance when the risky asset follows geometric frac tional Brownian motion. According to the principle of maximizing the investor＇ s expected lifetime utility, the optimal portfolio and consumption model with insurance is constructed by using Bellman dynamic programming principle. Then, the closed-form solutions for the optimal portfolio, insurance and consumption are derived under the condition that both the special consumption utility function and scrap function are given. Furthermore, some important properties for the influence of parametric changes on the optimal portfolio are also obtained. Finally, numerical examples are presented to discuss the change over the optimal portfolio and insurance and maximal utility in varying time intervals and Hurst exponent.