A Game Between Two Insurance Companies With Jump-diffusion Risk Model

In the modern insurance business,as the competition is intense,investment and reinsurance are essential for the insurance company to share risks and enlarge the income.At present,most of the optimal investment and reinsurance strategy are given in the objective of maximizing the expected utility or minimizing the ruin probability.There are few articles considering the game between the insurance companies.It is still in the preliminary stage of development.Most relevant literatures consider investment and reinsurance in the diffusion approximation risk model instead of jumpdiffusion model.That is because it is hard to get analytic solution with complex calculation.Jin considers a stochastic differential reinsurance game between two insurance companies by a mixed regime-switching jump-diffusion model and gives the numerical solution of the optimal reinsurance strategy.Based on such reality,the paper combines jump-diffusion model and game theory,considering a non-zero reinsurance game between two insurance companies under jump-diffusion model.In this paper,we assume that there are one risk-free assets(such as bonds)and one kind of risky assets(such as stock)available for insurance companies to invest.At the same time,this paper considers the optimal reinsurance problem with proportional reinsurance which is assumed to be calculated via expected premium principle.We establish the Hamilton–Jacobi–Bellman equations under the goal of maximizing the utility of the difference between the two insurance companies' terminal surplus,which is modeled by jump-diffusion risk process.We also prove the existence of Nash equilibrium between the two companies by applying the method of game theory and stochastic dynamic programming principle,and give the Nash equilibrium strategy.In some special cases,the influences of economic variables on our optimal strategies are demonstrated and some economic explanations are given accordingly.