A hybrid stochastic differential reinsurance and investment game with bounded memory

This paper investigates a hybrid stochastic differential reinsurance and investment game between one reinsurer and two insurers, including a stochastic Stackelberg differential subgame and a non-zero-sum stochastic differential subgame. The reinsurer, as the leader of the Stackelberg game, can price reinsurance premium and invest her wealth in a financial market that contains a risk-free asset and a risky asset which is described by the constant elasticity of variance (CEV) model. The two insurers, as the followers of the Stackelberg game, can purchase proportional reinsurance from the reinsurer and invest in the same financial market. The competitive relationship between two insurers is modeled by the non-zero-sum game, and their decision-making will consider the relative performance measured by the difference in their terminal wealth. The bounded memory feature is characterized by the wealth process with delay. The purpose of the reinsurer is to maximize the expected utility of her own terminal wealth with delay. The two insurers aim to maximize the expected utility of the combination of her terminal wealth and the relative performance with delay. By using the idea of backward induction and the dynamic programming approach, we derive the equilibrium strategy and value functions explicitly. Then, we provide the corresponding verification theorem. Finally, some numerical examples and sensitivity analysis are presented to demonstrate the effects of model parameters on the equilibrium strategy. We find the delay factor discourages or stimulates investment depending on the length of delay. Moreover, competitive factors between two insurers make their optimal reinsurance-investment strategy interact, and reduce reinsurance demand and reinsurance premium price.