Optimal multidimensional reinsurance policies under a common shock dependency structure

In this paper, we consider an insurance company that is active in multiple dependent lines. We assume that the risk process in each line is a Cramér–Lundberg process. We use a common shock dependency structure to consider the possibility of simultaneous claims in different lines. According to a vector of reinsurance strategies, the insurer transfers some part of its risk to a reinsurance company. Our goal is to maximize our objective function (expected discounted surplus level integrated over time) using a dynamic programming method. The optimal objective function (value function) is characterized as the unique solution of the corresponding Hamilton–Jacobi–Bellman equation with some boundary conditions. Moreover, an algorithm is proposed to numerically obtain the optimal solution of the objective function, which corresponds to the optimal reinsurance strategies.