Time-Inconsistent Singular Control Problems with a Running Minimum Process

TL;DR

This paper introduces a novel time-inconsistent singular control framework with a running minimum, establishing theoretical foundations, existence of equilibria, and demonstrating practical implications through a dividend problem example.

Contribution

It develops a weaker regularity verification theorem, characterizes admissible controls, and proves equilibrium existence in a complex, path-dependent setting.

Findings

Existence and uniqueness of solutions to Skorokhod reflection problems.

Monotonicity and local concavity of the dividend boundary.

Numerical simulations confirm equilibrium robustness.

Abstract

This paper develops a time-inconsistent and path-dependent singular control framework incorporating a running minimum process. We derive a verification theorem that characterizes equilibria under substantially weaker regularity conditions than those imposed in the existing literature, and we obtain a stronger notion of equilibrium by enlarging the class of feasible perturbations. We first establish the mathematical foundations of the framework by proving the existence and uniqueness of strong solutions to a class of Skorokhod reflection problems involving the running minimum and by characterizing admissible singular control laws. We further demonstrate the existence of an equilibrium through a dividend problem, where the running minimum leads to a highly coupled and nonlinear differential-algebraic system. For this problem, we prove the monotonicity and local concavity of the dividend boundary, thereby providing a mathematical explanation for dividend smoothing and scarring effects. Numerical simulations confirm the robustness of the equilibrium across a wide range of parameter values.