A multi-layer stochastic differential investment and reinsurance game with a dynamic contagion claim model

This paper studied the optimal investment and reinsurance problems between $ n $ insurers and two reinsurers within a multi-layer game framework. Specifically, the Stackelberg differential games were employed to describe the game relationship between the insurers and the reinsurers. Furthermore, a non-zero-sum game model was established by modeling the competitive relationship among insurers. Considering the clustering nature of claim occurrences, the diffusion approximation of the compound Hawkes process was employed to model the claim process. It was assumed that investors can allocate funds to a risk-free asset and different risky assets, where the volatility of the risky asset is characterized by the bi-fractional Brownian motion. By optimizing the expected utility of terminal wealth as the criterion, employing dynamic programming and game theory, equilibrium investment and reinsurance strategies were derived. Finally, through numerical analysis, the impact of relevant parameters on equilibrium strategies was explored. The results suggest that when there is fierce competition among reinsurers, they will lower reinsurance premiums to attract insurers, which will allow insurers to purchase reinsurance at a lower cost.