Abstract: With the acceleration of the process of global economic integration, the cooperation among multiple countries, firms or individuals has become popular. In insurance market, many insurance companies often cooperate to jointly resist claim risk. Under the cooperation situation, each insurance company should consider both his own benefit and the benefits of others; otherwise, their cooperation will be destroyed. Based on such a reality, n insurance companies should take into account their joint benefits in the cooperation case. Based on this consideration, we establish a cooperation model for many insurance companies.
In the process of making reinsurance strategy, the insurance and reinsurance company often compete. There are at least three aspects of competition between them. Firstly, they compete in insurance business and reinsurance business. Since the reinsurance company may engage in insurance business, it will compete with the insurance company, and vice versa. Secondly, when signing a reinsurance contract, they have conflicting interests, and both parties want to maximize their own interests. Thirdly, after signing a reinsurance contract, the insurance company does not expect the reinsurance company's income, i.e., the reinsurance premium, to exceed his remaining income, i.e., the total premium minus reinsurance premium. Usually, the insurance company only pays a small amount of reinsurance premium, and then invests the remaining premium in the financial market. Therefore, we quantify the competition between n insurance companies and a reinsurance company by relative performance.
Based on the cooperation model and competition model, we propose a unified competition and cooperation framework, and consider the corresponding reinsurance problem. The main research goal of each insurance company is to find an optimal time-consistent reinsurance strategy so as to maximize the expected terminal wealth while minimizing the variance of the terminal wealth. By using stochastic calculus technology, a Hamilton-Jacobi-Bellman (HJB) equation and the corresponding verification theorem are established. Furthermore, we derive the explicit optimal reinsurance strategy for the considered reinsurance problem. We also present the optimal reinsurance strategies for three special cases: the case without considering the dependence between insurance businesses, the case without considering the cooperation, and the case without considering the competition. Finally, the influence of the key model parameters on the optimal time-consistent reinsurance strategy is analyzed by numerical experiments.
Through theoretical analysis and numerical experiments, we have the following new findings: (1)As the number of cooperative insurance companies increases, each insurance company will reduce his retention level of reinsurance. (2)With an increase in competition degree, each insurance company will increase his retention level of reinsurance. (3)With an increase in cooperation degree among n insurance companies, each insurance company will reduce his retention level of reinsurance. (4)With an increase in the dependence among n insurance companies, each insurance company will reduce his retention level of reinsurance. These results can effectively guide insurance companies to make more reasonable reinsurance decisions in the competitive and cooperative environment.
There are still some issues worthy of study in the future. First of all, it is interesting to apply our general framework and solutions to real insurance problems and to examine the proper selection of model parameters. Secondly, it is worthwhile to consider other forms of interaction among n insurance companies, although this is a very difficult job. Finally, we can also consider other objectives such as the general utility function or the mean-variance criterion with state dependent risk aversion parameter.

Key words: reinsurance, competition, cooperation, stochastic dynamic programming, stochastic control

摘要： 随着全球经济一体化进程的加快,多家保险公司经常通过合作共同抵御索赔风险。因此,我们建立了多家保险公司之间的合作模型。在再保险策略制定的过程中,保险公司和再保险公司经常进行竞争。因此,我们量化了n家保险公司与一家再保险公司之间的竞争,得到了兼有合作与竞争的模型,研究了由此导致的多家保险公司的最优再保险问题。每家保险公司的盈余过程满足复合泊松保险模型,每家保险公司通过购买比例再保险减少索赔风险。以均值-方差准则为目标,本文得到了竞争与合作情形下,最优再保险策略的解析解。最终,提供了数值实验,解释了理论结果的实际指导意义。

关键词: 再保险, 竞争, 合作, 随机动态规划, 随机控制