Abstract: In this paper, we combine a jump-diffusion model and the game theory, considering a non-zero reinsurance game between two insurance companies under a jump-diffusion model. 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 the 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 the stochastic dynamic programming principle, and give a 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.

Key words: reinsurance, jump-diffusion risk model, Hamilton-Jacobi-Bellman equations, sto-chastic differential game