On a class of non-zero-sum stochastic differential dividend games with regime switching

Abstract

This paper investigates a class of non-zero-sum stochastic differential game problems between two insurance companies. The surplus process of each company is modeled by a Brownian motion where drift and volatility depend on the continuous-time Markov regime switching process. Both companies have the option of paying dividends. The objective is to maximize the expected discount utility of surplus relative to a reference point for each insurer, the gains brought by the insurer’s own dividend payout, and the losses incurred by the dividend payment of the competitor. The gains and losses are proportional to the amount of corresponding dividend payment. To find the optimal dividend policy, we relate this singular control problem to a stopping game. Further, we prove the link based on the verification theorem and show that value functions of the non-zero-sum stochastic differential problem can be derived by integrating value functions of a stopping game. Finally, we apply our results to the case with two regimes and the case without regime switching. Numerical examples are also provided.