Abstract: There is keen competition between insurersin the financial markets. Insurers not only seek to maximize their own wealth, but also focus on comparing their wealth with that of their competitors. This paperstudies a class of stochastic differential investment and proportional reinsurance game with delay, and investigates the effects of delay and competition coefficients on optimal investment and reinsurance strategies of insurers. Firstly, for the classical Cramér-Lundberg model, equilibrium investment and reinsurance strategies with delay are obtained using the differential game and stochastic optimal control theory, and an explicitform of the value function is given. Secondly, further for the approximate diffusion process, explicit expressions for the equilibrium investment strategy and proportional reinsurance strategy under exponential utility are obtained. The sensitivity of equilibrium strategy to the model parameters is analyzed by a numerical example.The results show that: whether or not the delay information is taken into account will have a significant impact on the final equilibrium reinsurance strategy and optimal investment strategy. The more value of wealth at an earlier time is considered, the more prudent and rational insurers' investmentbehavior will be; on the other hand, the more importance they place on industry competition, the more risky andaggressivetheir investment behaviortends to be.

Key words: delay, investment and proportional reinsurance, nash equilibrium, stochastic differential game

摘要： 金融市场不断发展,激烈的市场竞争使得相对绩效比较在保险机构的业绩评估中占据越来越重要的地位。考虑历史业绩对公司决策的影响,引入时滞效应,研究时滞效应对具有竞争关系公司之间最优投资策略和最优再保险策略的影响。运用随机最优控制和微分博弈理论,针对Cramér-Lundberg模型,得到了均衡投资和再保险策略,给出了值函数的显式解;然后进一步针对近似扩散过程,求得指数效用下均衡投资策略和比例再保险策略的显式表达。通过数值算例,分析了最优均衡策略随模型各重要参数的动态变化。结论显示:保险公司在决策时是否将时滞信息纳入考虑之中将大大影响其投资和再保险行为。保险公司考虑较早时间财富值越多,其投资再保险行为就表现得越趋向于保守和谨慎;与之相反,如果保险公司对行业间的竞争越看重,其投资再保险策略就越倾向于冒险和激进。

关键词: 时滞效应, 投资与比例再保险, Nash均衡, 随机微分博弈