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

In Bai and Paulsen (SIAM J. Control optim. 48, 2010) 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 $\bar{u}^*$, reduce it to $\bar{u}^*-\xi$ through a dividend payment. After transaction costs, the shareholder receives $k\xi-K$. 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.