SAMS Subject Classification Number: 25

This talk presents a stochastic differential game between two insurance companies, a big one and a small one. The distinctions of these companies is that the big company has sufficient asset to invest in a risk-free asset and a risky asset and is allowed to purchase proportional reinsurance or acquire new business, and the small company can transfer part of the risk to a reinsurer via proportional reinsurance. The game studied here is zero-sum, where the big company is trying to maximize the expected exponential utility of the difference between two insurance companies’ surpluses at the terminal time to keep its advantage on surplus, while simultaneously the small company is trying to minimize the same quantity to reduce its disadvantage. The relationships between the surplus processes and the price process of the risky asset are considered. The Nash equilibrium strategy is obtained through verification theorem which rest on the stochastic control theory.