Optimal Dividend Payments in a Dual Risk Model with both Fixed and Proportional Costs

In this paper, we study the optimal dividend problem in a dual risk model, which might be appropriate for companies that have fixed expenses and occasional profits. Assuming that dividend payments are subject to both proportional and fixed transaction costs, our object is to maximize the expected present value of dividend payments until ruin, which is defined as the first time the company's surplus becomes negative. This optimization problem is formulated as a stochastic impulse control problem. By solving the corresponding quasi-variational inequality (QVI), we obtain the analytical solutions of the value function and its corresponding optimal dividend strategy when jump sizes are exponentially distributed.