Time-Consistent Investment-Reinsurance Strategy for Mean-Variance Insurers With a Defaultable Security

This paper considers an optimal investment and reinsurance problem involving a defaultable security for an insurer under the mean-variance criterion in a jump-diffusion risk model. The insurer is allowed to purchase proportional reinsurance or acquire new business and invest in a financial market consisting of a risk-free bank account, a stock and a defaultable bond. From a game theoretic perspective, the extended Hamilton-Jacobi-Bellman systems are established for the post-default case and the pre-default case, respectively. Furthermore, for the two cases, closed-form expressions for the optimal time-consistent investment-reinsurance strategies and the corresponding optimal value functions are derived, and some properties of the strategies are analyzed. Finally, some special cases of our model are presented, and numerical analysis is provided to illustrate our results