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

TL;DR

This paper addresses an optimal dividend problem under partial information, formulating it as a singular stochastic control with absorption in a degenerate diffusion, and introduces a novel probabilistic approach linking control and stopping problems.

Contribution

It develops a new probabilistic method to solve a complex singular control problem with absorption, establishing a connection to optimal stopping with creation at a local-time dependent rate.

Findings

Value function is a smooth solution to the free boundary problem.

Constructed an explicit optimal dividend strategy.

Linked multidimensional control with stopping problems involving creation.

Abstract

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 2-dimensional degenerate diffusion, whose first component is singularly controlled 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 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 2-dimensional reflecting diffusion.