Free boundary regularity for optimal dividend problems

Problem's Description

The free boundary regularity for singular control problems is important in order to be able to construct an optimal control (given by a reflection of the stochastic process). For so called optimal dividend problems (a class of problems on optimal corporate capital structure), this is characterized by the interface between an elliptic condition and a gradient constraint in one direction.

In particular, without any other control variables available, if $x$ describes the current capital/cash, $r$ is the discount rate, and $\mathcal{A}$ is the infinitesimal generator for the state processes, then $$ \min \{ (r - \mathcal{A}) V, V_x -1\} = 0, $$ is the HJB equation of interest, interpreted in the viscosity sense. The precise domain and boundary conditions depend on modeling choices.

To be concrete, in the referenced paper, where the profitability of the firm is random and given by an Ornstein–Uhlenbeck process, the equation is $$ \min \left\{ rV - \mu V_x - k(\mu - \bar{\mu}) V_\mu - \frac{\sigma^2}{2} V_{xx} - \frac{\tilde \sigma^2}{2} V_{\mu\mu} - \rho \sigma \tilde\sigma V_{x\mu}, V_x - 1 \right\} = 0, \quad (x,\mu) \in [0,\infty)\times \mathbb{R},$$ with strictly positive parameters and the boundary condition $V(0, \mu) = 0$. Note that we do not expect this equation to have a classical solution due to what we expect to be a discontinuity in $V_{\mu\mu}$ along one section of the free boundary.

The free boundary regularity (and shape) of this PDE is important to better understand the control problem. The question is also of interest for a large class of related problems, with the common denominator that they share the $V_x - 1$ constraint.

1. Article Optimal dividend policies with random profitability

   • Max Reppen,
   • Jean-Charles Rochet,
   • Halil Mete Mete Soner

   arXivfulltext