Investment And Reinsurance Strategies With Stochastic Differential Games

In this article, we consider a stochastic optimal control problem of two insurance companies in continuous time. Each of them can employ strategies including invest-ment and reinsurance. We make a assumption that two companies are faced with different insurance risks, which may be correlative. And one can invest in a risk asset, which is not the same with another. Both companies can invest in a same risk-free asset. A company’s wealth process can be described as a stochastic differential equa-tion. In stochastic optimal control problem in continue time, one takes strategies to make his payoffs maximum. Combining the two processes of two companies to form one, we establish a single payoff function for two companies. One always wants to maximize it, while another acts oppositely. A Zero-sum stochastic differential game can describe this problem perfectly. In our model, we let investment amount and re-tention proportion be strategies. We assume that one can borrow money to invest by a limit but can not short sell the stock. And retention proportion is from0to1. Con-sidering discount rate, we describe the Nash equilibrium. Given a special form of the payoff function, we study a probability maximizing game. By analyzing the domain of the value function, we obtain the explicit solutions of strategies. At last a numeral case is been showed.