关键词: 再保险, 竞争, 合作, 随机动态规划, 随机控制

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