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arXiv · 2605.19914

Time-Inconsistent Singular Control Problems with a Running Minimum Process

Rui Dai·Guohui Guan·Zongxia Liang·Xiaodong Luo
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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.

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Rui Dai, Guohui Guan, Zongxia Liang, Xiaodong Luo. 2026-05-19. Time-Inconsistent Singular Control Problems with a Running Minimum Process. https://arxiv.org/abs/2605.19914

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