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

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