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

Tiziano De Angelis, Erik Ekström, Marcus Olofsson

Published 2022-06-23Version 1

Motivated by a formulation of the classical dividend problem we develop a maximality principle for singular stochastic control problems with 2-dimensional degenerate dynamics and absorption along the diagonal of the state space. This result is new in the theory of singular control and it unveils deep connections to Peskir's maximality principle in optimal stopping (Ann. Probab. 26, no. 4, 1998). We construct an optimal control as a Skorokhod reflection along a moving barrier. The barrier can be computed analytically as the smallest solution to a certain class of ordinary differential equations (ODEs). Contrarily to the classical 1-dimensional formulation of the dividend problem our framework produces a non-trivial solution when the firm's capital evolves as a geometric Brownian motion (gBm). Such solution is also qualitatively different from the one traditionally obtained for the arithmetic Brownian motion.