扩散风险模型中的最优投资、分红与再保险策略研究

To spread risks and improve safety, insurance companies need reinsurance, to increase profits they need investment, and to ensure the shareholders’ interests they offer dividends . Thus, to make strategies on the optimal investment, reinsurance and dividends under different objective functions is a key issue concerning an insurance company’s safety, profits and shareholders’ interests, and a question about proposed stochastic optimal control. For different objective functions, different risk models, even various constrains, there are different optimal strategies or strategy combinations which would even be completely different. Most former objective functions aims to maximize the shareholders’ cumulative total dividend discount or to make the longest survival for a company. This subject would consider the two mentioned above by introducing the comprehensive optimization objective function. Under comprehensive objects, this paper first discusses the optimal dividends and investment strategies in diffusion models, second discusses the optimal excess reinsurance and investment strategies , finally，in objective functions with minimax principle, control by adopting the Robust Optimal determines optimal investments, dividends and reinsurance strategies of insurance companies to which there is ambiguity aversion .

对保险公司来说，为了分散风险以提高安全性，需要再保险；为了增加盈利，需要投资；为了股东利益，需要给股东分红。因而在不同的目标函数下，决定最优分红、再保险和投资策略，关乎公司的安全性、盈利能力和股东利益，都是相当核心的问题，并且是一个随机最优控制问题。对于不同的目标函数、不同的风险模型、甚至是所施加的各种约束，所对应的最优策略或策略组合肯定是有区别，甚至是完全不同的。以往的目标函数，大多是使得股东的累计分红折现总量最大或者是使得公司的生存时间最长。在本课题中我们将兼顾两者，引入综合优化目标函数，在综合目标下，先探讨扩散风险模型中的最优分红和最优投资策略；再探讨最优超额再保险和最优分红策略；最后针对模糊厌恶的保险公司，在极大极小原则的目标函数下，利用鲁棒最优控制原理，求出公司的最优投资、分红和再保险策略。

考虑了一个具有违约风险的鲁棒最优投资再保险问题。保险公司(AAI)可以进行无风险资产、股票和可违约公司债券的交易。股票的价格用一个跳跃扩散过程来描述，跳跃强度和跳跃幅度分布都是不确定的，即跳跃为模糊的。假设 AAI的盈余过程遵循近似扩散过程，特别是再保险保费按广义均值-方差保费原则计算，再保险类型必须遵循.self-reinsurance函数。在执行动态规划时，对预默认情况和后默认情况都进行分析，在最坏情况下，给出了最优策略和相应的值函数。此外，我们给出了一个验证定理进行了详细的证明，并给出了一些特例和数值例子来说明我们的理论结果。