Optimal reinsurance with both proportional and fixed costs

This paper investigates an optimal reinsurance problem with both proportional and fixed transaction costs. Associated with a reinsurance, the proportional cost is usually charged by the reinsurer and the fixed cost occurs at the beginning of the reinsurance due to consultant commission. We describe an insurer’s original cash reserve process by a Brownian motion with positive drift. The insurer aims to minimize its ruin probability by taking a dynamic reinsurance strategy. This leads to a mixed regular control and optimal stopping problem. We solve it by establishing a connection with an optimal stopping problem. The value function and the optimal reinsurance strategy are obtained explicitly.