Non-zero sum stochastic differential game between two insurance companies under the variance premium principle

Abstract

Abstract This paper considers a non-zero-sum stochastic differential game between two competitive companies where both companies aim to seek the optimal investment and reinsurance strategies. Furthermore, these two companies are allowed to purchase proportional reinsurance to mitigate the company's claim risks and, at the same time, invest in one risk-free asset and one risky asset whose price dynamics follow the geometric Brownian motion and take dividend payments into account. In particular, under the proportional reinsurance contract, the reinsurance premium is assumed to be calculated via the variance value principle instead of the expected value principle. The optimal reinsurance and investment strategies of both companies can be determined by solving the Fleming-Bellman-Isaacs equations for an exponential utility function. In the end, several numerical simulations were conducted to show how model parameters affect equilibrium reinsurance and investment strategies, and some conclusions about economics from these findings were deduced. JEL Classification: C73 , D53 , G11 , G22 MSC Classification: 91A15 , 93E20 , 91G15 , 97M30