Stochastic differential game strategies in the presence of reinsurance and dividend payout

Abstract

<p style='text-indent:20px;'>This paper presents and examines a problem in which two insurance companies apply non-proportional reinsurance to control risk. Additionally, each firm pays out dividends. The situation is modelled as a zero-sum stochastic differential game between the two companies. The goal of one company is to maintain business competitive advantage over the other by sustaining or increasing the difference between the respective liquid reserves of the two companies while the second company aims to minimise that difference. A verification theorem is formulated, proved and subsequently employed to derive the saddle point components. For the case of the payoff with a non-zero running cost function, we are able to solve explicitly the differential game. Numerical simulations are presented to illustrate the results as well as the economic interpretation.</p>