Optimal reinsurance-investment problem for two competitive or cooperative insurers under two investment patterns

The optimal reinsurance-investment problem for two competitive or cooperative insurers is investigated. Each insurer transfers part of the claims risk via reinsurance and invests the surplus in a financial market consisting of one risk-free asset and two risky assets for increasing wealth. Both the aggregated claims of the two surplus processes and the aggregated jumps of the two risky assets are governed by a common Poisson process. We consider two relationships between the two insurers, i.e., competition and cooperation. In addition, we consider two kinds of investment patterns, called the concentration investment and the diversification investment. The objective of the each insurer is to choose the optimal reinsurance-investment strategy so as to maximize the expected terminal wealth while minimizing the variance of the terminal wealth. By the technique of stochastic control theory, the explicit optimal reinsurance-investment strategies and the optimal value functions are obtained in the two investment patterns for the competition case and the cooperation case, and some special cases. Finally, some numerical experiments are carried out to illustrate the effects of model parameters on the optimal strategies and the optimal value functions, which reveal some interesting phenomena and provide useful guidances for reinsurance and investment in reality.