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具有模型不确定性的最优投资和再保险策略
【作者】 桂有利；
【导师】 林祥；
【作者基本信息】 中南大学 ， 概率论与数理统计， 2009， 硕士
【摘要】 近年来,由于保险行业竞争激烈,保险公司一方面通过对公司盈余进行投资,从投资中获得大量的收益来提高自己的偿付能力,同时保险公司为了减少自身所面临的大赔付的风险,又必须对赔付进行再保险处理。控制投资和再保险,使得期望财富效用最大或破产概率最小,无论在理论上,还是在保险实务中,都具有十分重要的意义。本文首先对进行停止损失再保险的传统风险模型做扩散逼近,使模型简单化,以最小破产概率为目标函数,借助HJB方程,求得了最优自留额满足的表达式,在个体理赔取指数分布和Erlang(2)分布的情况下求出了最优自留额和最小破产概率的显示解,并通过数值计算得到自留额与各参数之间的关系。其次,对不确定情况下的跳-扩散风险最优投资和再保险进行研究。在保险公司的盈余和投资具有不确定性的条件下,得到了使得期望指数效用最大的最优投资、再保险和概率测度变换,以及最优值函数的显示表达式。
【Abstract】 In recent year, due to severe competition in the insurance industry, the insurance companies get a lot of benefits to improve their solvency from the investment of the money at its disposal. At the same time, in order to reduce the risk of large claims, the insurance company takes reinsurance policy for claims. Subject to the control of investment and reinsurance, maximizing the expected wealth utility and minimizing the probability of ruin have a great deal of theoretical and practical significance.Firstly, the paper studies the diffusion approximation of the classical risk model. The HJB equation of the minimal probability of ruin and the optimal stop-loss reinsurance is given. We obtain the explicit expression of the minimal probability of ruin and the optimal stop-loss reinsurance under the claim is exponential distribution and Erlang distribution. The effects of the parameters on the optimal retention level are obtained by numerical calculation. Secondly, we study the optimal investment and reinsurance in the jump-diffusion risk model under uncertainty. Under uncertainty of the surplus and investment, explicit expressions for the maximal expected exponential utility and the corresponding optimal policies are obtained.