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

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 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.

Optimal reduction of public debt under partial observation of the economic growth

“…To take care of the second derivative V yy we follow ideas used in De Angelis [19]. In particular, we determine the second weak derivative of V (recall that V y is continuous by Theorem 4.9) and then show that it is a continuous function.…”

“…Theoretical results on the PDE characterisation of the value function of a two-dimensional optimal correction problem under partial observation are obtained by Menaldi and Robin [53], whereas a general maximum principle for a not necessarily Markovian singular stochastic control problem under partial information has more recently been derived by Øksendal and Sulem [54]. We also refer to De Angelis [19] and Decámps and Villeneuve [23] who provide a thorough study of the optimal dividend strategy in models in which the surplus process evolves as a drifted Brownian motion with unknown drift that can take only two constant values, with given probabilities.…”

Two-sided Singular Control of an Inventory with Unknown Demand Trend

“…Such an uncertainty about the drift therefore adds a structural risk component to decision making, in addition to the noise from the stochastic driver of the underlying process. Such scenaria have already received attention in the mathematical economic/financial literature, such as [13] for investment timing, [8] for asset trading, [18] for optimal liquidation, [16] for contract theory, and [12] and [14] for dividend payments.…”

“…However, the literature on the characterisation of the optimal policy in singular stochastic control problems with partial observation is limited, and actually deals only with monotone controls. We firstly refer to [33] that studies singular control problems with partial information via the study of their associated backward stochastic differential equations (BSDEs) leading to general maximum principles; [12] that solves the optimal dividend problem under partial information on the drift of the revenue process of a firm that can default, creating also an absorption state; [14] that studies a dynamic model of a firm whose shareholders learn about its profitability, face costs of external financing and costs of holding cash; and [4] that considers the debt-reduction problem of a government that has partial information on the underlying business conditions. Contrary to the aforementioned papers with monotone controllers, we allow the decision maker to both decrease and increase the underlying process by using controls of bounded-variation.…”

“…Using these properties, and proving local semiconcavity of V , allow us to show via fine techniques from viscosity theory that V ∈ C 1 (R × (0, ∞)). Because of the degeneracy of the process (X, Φ) (in which X and Φ are driven by the same Brownian motion), in order to derive further regularity of the control problem's value function it is useful to derive the intrinsic parabolic formulation of the problem (see also [25] and [12]). This is achieved by passing yet to another transformation (X, Y ) of our state process taking values in R 2 .…”