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

中文翻译：

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具有部分信息的最优红利并停止退化的反射扩散

我们为拥有部分获利能力信息的公司经理研究最优股息问题。该问题被表述为具有随机变量的奇异随机控制之一，其中包含有关潜在过程漂移和吸收的部分信息。在马尔可夫公式中，我们有一个二维简并扩散，其第一个分量受到单个控制。而且，该过程在其第一部分达到零时被吸收。由于简并和吸收的相互作用，从分析的角度来看，与控制问题的价值函数相关的自由边界问题（FBP）具有挑战性。我们找到一种概率方法来证明分红问题的价值函数是FBP的平滑解，并构造了最优分红策略。我们的方法在具有吸收的多维奇异随机控制问题与“创造”的最佳停止问题之间建立了新的联系。停止问题的一个关键特征是，创建以辅助二维反射扩散的“本地时间”的状态相关速率发生。