Stochastic differential game formulation on the reinsurance and investment problem

This paper focuses on a stochastic differential game between two insurance companies, a big one and a small one. 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 maximise 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 minimise the same quantity to reduce its disadvantage. Particularly, the relationships between the surplus processes and the price process of the risky asset are considered. By applying stochastic control theory, we provide and prove the verification theorem and obtain the Nash equilibrium strategy of the game, explicitly. Furthermore, numerical simulations are presented to illustrate the effects of parameters on the equilibrium strategy as well as the economic meanings behind.