Impulse control for the optimal dividend and reinvestment problem

In this paper, we mainly consider the dividend payments and reinvestments with both fixed and proportional transaction costs in the diffusion model. Because we have taken into account the fixed transaction costs, the problem was changed the impulse stochastic control problem. The final goal is to maximize the expected value of discounted net cash flow, i.e. dividends paid minus reinvestments,it makes shareholders get the most profits. We find the quasi-variational inequalities for the value function, and consider properties of the value function. We identify the optimal policy and the value function. In the end, we give the relationship between dividend payments, reinvestments and transaction costs.