A Stackelberg-Nash equilibrium with investment and reinsurance in mixed leadership game

In this paper, we investigate the optimal reinsurance and investment problem from joint interests of the insurer and the reinsurer in the framework of the mixed leadership game, where both of them can invest their wealth in the financial market, and the liabilities are correlated with the risky asset. More specifically, we assume that the insurer is the leader to determine the amount he/she invests into the risky asset but acts as the follower to decide the optimal reinsurance retention level, while the reinsurer is the leader to set the reinsurance premium price and acts as a follower to determine his/her investment strategy. Both the insurer and the reinsurer aim to maximize the expected utility of their terminal wealth, and the explicit Stackelberg-Nash equilibrium is derived by solving the corresponding Hamilton-Jacobi-Bellman (HJB) equations. In order to better understand the effect of the mixed leadership game on the optimal reinsurance and investment problem, we also consider the optimization problem within the framework of the stochastic Stackelberg differential game shown in [Bai, Y., Zhou, Z., Xiao, H., Gao, R. & Zhong, F. (2021). A Stackelberg reinsurance-investment game with asymmetric information and delay. Optimization 70(10), 2131-2168.], where the reinsurer is the leader while the insurer acts as the follower for both the reinsurance and investment strategies. Some theoretical analyses and numerical examples are provided to show the economic intuition and insights of the results.