A stochastic Stackelberg differential reinsurance and investment game with delay in a defaultable market

In this paper, we investigate a stochastic Stackelberg differential reinsurance and investment game problem with delay for a reinsurer and an insurer in a defaultable market, which consists of a risk-free asset, a risky asset and a defaultable bond. As the leader, the reinsurer can determine reinsurance premium price and investment strategy to maximize the expected exponential utility of its terminal wealth with delay. As the follower, the insurer can select reinsurance proportion and investment strategy to maximize the expected exponential utility of its terminal wealth with delay. By using the idea of backward induction and the dynamic programming approach, we solve the leader's and follower's optimization problems sequentially and derive the Stackelberg equilibrium strategy explicitly. Then, we provide the corresponding verification theorem. Finally, we present some numerical examples to illustrate the influence of model parameters on the equilibrium strategy and draw some economic interpretations from these results. We find that the pre-default value functions are higher than the post-default value functions and the influence of delay weight on equilibrium strategy depends on the length of delay time. Moreover, when the Stackelberg equilibrium is achieved in the interior case, the optimal reinsurance premium follows the variance premium principle and the influence of delay weight on the optimal reinsurance premium strategy is just opposite to that on other strategies.