The maximality principle in singular control with absorption and its applications to the dividend problem

TL;DR

This paper extends Peskir's maximality principle to complex singular control problems with absorption, providing explicit solutions for a generalized dividend problem involving geometric Brownian motion.

Contribution

It introduces a novel application of the maximality principle to 2D degenerate control problems with absorption, deriving explicit barrier-based solutions.

Findings

Optimal control as Skorokhod reflection along a moving barrier

Explicit analytical computation of the barrier via a non-linear ODE

New solutions for dividend problems with geometric Brownian motion

Abstract

Motivated by a new formulation of the classical dividend problem, we show that Peskir's maximality principle can be transferred to singular stochastic control problems with 2-dimensional degenerate dynamics and absorption along the diagonal of the state space. We construct an optimal control as a Skorokhod reflection along a moving barrier, where the barrier can be computed analytically as the smallest solution to a certain non-linear ordinary differential equation. Contrarily to the classical 1-dimensional formulation of the dividend problem, our framework produces a non-trivial solution when the firm's (pre-dividend) equity capital evolves as a geometric Brownian motion. Such solution is also qualitatively different from the one traditionally obtained for the arithmetic Brownian motion.