Optimal Dividend, Reinsurance, and Capital Injection Strategies for an Insurer with Two Collaborating Business Lines

Abstract: This paper considers an insurer with two collaborating business lines, and the risk exposure of each line follows a diffusion risk model. The manager of the insurer makes three decisions for each line: (i) dividend payout, (ii) (proportional) reinsurance coverage, and (iii) capital injection (from one line into the other). The manager seeks an optimal dividend, reinsurance, and capital injection strategy to maximize the expected weighted sum of the total dividend payments until the first ruin. We completely solve this problem and obtain the value function and optimal strategies in closed form. We show that the optimal dividend strategy is a threshold strategy, and the more important line always has a lower threshold to pay dividends. The optimal proportion of risk ceded to the reinsurer is decreasing with respect to the aggregate reserve level for each line, and capital injection is only used to prevent the ruin of a business line. Finally, numerical examples are presented to illustrate the impact of model parameters on the optimal strategies.