Optimal dividends with partial information and stopping of a degenerate reflecting diffusion

We study the optimal dividend problem for a firm's manager who has partial on the profitability of the firm. The problem is formulated as one singular stochastic control with partial information on the drift of the process and with absorption. In the Markovian formulation, we have a2-dimensional degenerate diffusion, whose first component is singularly and it is absorbed as it hits zero. The free boundary problem (FBP) to the value function of the control problem is challenging from the point of view due to the interplay of degeneracy and absorption. We a probabilistic way to show that the value function of the dividend is a smooth solution of the FBP and to construct an optimal dividend. Our approach establishes a new link between multidimensional singular control problems with absorption and problems of optimal stopping `creation'. One key feature of the stopping problem is that creation at a state-dependent rate of the `local-time' of an auxiliary2-dimensional reflecting diffusion.