A hybrid reinsurance-investment game with delay and asymmetric information

In this paper, we investigate the optimal reinsurance and investment problem in the framework of the hybrid stochastic differential game, which includes a stochastic Stackelberg differential subgame and a non-zero-sum stochastic differential subgame. The stochastic Stackelberg differential subgame considers the leader–follower relationship between a reinsurer and two insurers. The reinsurer, as the leader, prices reinsurance premium and invests its wealth in a financial market that contains a risk-free asset and a risky asset whose price process is described by the constant elasticity of variance (CEV) model. On the other side, the two insurers, as the followers, purchase proportional reinsurance from the reinsurer and invest in the same financial market. Also, the competitive relationship between the two insurers is characterized by the non-zero-sum stochastic differential subgame in which each insurer considers the relative performance. By introducing the delay feature into the hybrid game, we are able to obtain the wealth processes depicted by the stochastic delay differential equations. Considering the effect of asymmetric information, we assume the reinsurer and insurers have different levels of information about the financial market. The objective of the reinsurer is to maximize the mean–variance functional of its terminal surplus with delay, while each insurer is to maximize the mean–variance functional of its terminal surplus with delay relative to that of its competitor. By applying techniques in stochastic control theory, we obtain the extended Hamilton–Jacobi–Bellman equations, and then by using the idea of backward induction, we establish the equilibrium strategies and the corresponding value functions. Finally, we include some numerical examples and sensitivity analysis to illustrate the effects of model parameters on the equilibrium strategies.