On non-trivial barrier solutions of the dividend problem for a diffusion under constant and proportional transaction costs

In Bai and Paulsen [L. Bai, J. Paulsen, Optimal dividend policies with transaction costs for a class of diffusion processes, SIAM J. Control Optim. 48 (2010) 4987–5008] the optimal dividend problem under transaction costs was analyzed for a rather general class of diffusion processes. It was divided into several subclasses, and for the majority of subclasses the optimal policy is a simple barrier policy; whenever the process hits an upper barrier ū∗, reduce it to ū∗−ξ through a dividend payment. After transaction costs, the shareholder receives kξ−Kkξ−K.It was proved that a simple barrier strategy is not always optimal, and here these more difficult cases are solved. The optimal solutions are rather complicated, but interesting.