The Maximality Principle in Singular Control with Absorption and Its Applications to the Dividend Problem

SIAM Journal on Control and Optimization, Volume 62, Issue 1, Page 91-117, February 2024.
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 two-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 nonlinear ODE. Contrarily to the classical one-dimensional formulation of the dividend problem, our framework produces a nontrivial solution when the firm’s (predividend) equity capital evolves as a geometric Brownian motion. Such a solution is also qualitatively different from the one traditionally obtained for the arithmetic Brownian motion.