Optimal dividend and capital injection problem in the dual model with proportional and fixed transaction costs

In this paper we consider the dividend payments and capital injections control problem in a dual risk model. Such a model might be appropriate for a company that specializes in inventions and discoveries, which pays costs continuously and has occasional profits. The objective is to maximize the expected present value of the dividends minus the discounted costs of capital injections. This paper can be considered as an extension of , we include fixed transaction costs incurred by capital injections in this paper. This leads to an impulse control problem. Using the techniques of quasi‐variational inequalities (QVI), this optimal control problem is solved. Numerical solutions are provided to illustrate the idea and methodologies, and some interesting economic insights are included.