Optimal Monotone Mean-Variance Problem in Catastrophe Insurance Model

This paper considers an optimal investment and reinsurance problem with catastrophic risk. The insurer takes both the ordinary and the catastrophe insurance businesses. We assume that the occurrence of the catastrophe follows a Poisson process and influences the ordinary insurance business. The claim intensity of the ordinary insurance business is modelled by a Cox process with a short-noise intensity, which jumps upwards whenever a catastrophe happens and then decays as time goes by. The size of the catastrophe claim is assumed to be proportional to the size of the short- noise process. The aim is to maximize the monotone mean-variance(MMV) criterion, which is an amended version of the classical mean-variance criterion to guarantee its monotonicity. We formulate the original optimal MMV problem as an auxiliary zero-sum game. Then, by applying the dynamic programming approach and by solving the Hamilton-Jacobi-Bellman-Isaacs(HJBI) equation, we find the value function and the optimal strategy explicitly and also derive the efficient frontier under the MMV criterion