Optimal dividend and risk control policies in the presence of a fixed transaction cost

Abstract

In this paper, we consider a large insurance company whose cumulative cash flow process is described by a drifted Brownian motion. The preference of the insurer is to maximize his/her firm’s value, which corresponds to the expected present value of the dividend payments up to the ruin time. In the business process, the insurer has the option to draw up a dividend payment policy and to purchase proportional reinsurance at a certain point in time. In view of some typical practical expenses (e.g., consultant commission), it is assumed that a fixed transaction cost occurs at the beginning of the reinsurance commitment which, once made, is irreversible. This leads to a mixed stochastic control problem of optimal stopping time and singular control. For this mixed problem, we derive closed-form solutions for the optimal time to purchase reinsurance, the optimal retained risk proportion, the optimal dividend barrier, and the value function. The optimal solution shows that reinsurance is valueless to the insurer when the fixed cost is larger than a threshold, and comes into play when the fixed cost is less than this threshold. We also perform some numerical calculations to assess the impacts of fixed costs on the value function and the optimal policies.