Optimal reinsurance with multivariate risks and dependence uncertainty

In this paper, we study the optimal reinsurance design from the perspective of an insurer with multiple lines of business, where the reinsurance is purchased by the insurer for each line of business, respectively. For the risk vector generated by the multiple lines of business, we suppose that the marginal distributions are fixed, but the dependence structure between these risks is unknown. Due to the unknown dependence structure, the optimal strategy is investigated for the worst-case scenario. We consider two types of risk measures: Value-at-Risk ($\mathrm{VaR}$) and Range-Value-at-Risk (RVaR) including Expected Shortfall ($\mathrm{ES}$) as a special case, and general premium principles satisfying certain conditions. To be more practical, the minimization of the total risk is conducted under some budget constraint. For the $\mathrm{VaR}$-based model with only two risks, it turns out that the limited stop-loss reinsurance treaty is optimal for each line of business. For the model with more than two risks, we obtain two types of optimal reinsurance strategies if the marginals have convex or concave distributions on their tail parts by constraining the ceded loss functions to be convex or concave. Moreover, as a special case, the optimal quota-share reinsurance with dependence uncertainty has been studied. Finally, after applying our findings to two risks, some studies have been implemented to obtain both the analytical and numerical optimal reinsurance policies.