Time-inconsistent singular control problems: Reflection and Absolutely continuous controls with exploding rates

Abstract

We study a time-inconsistent singular stochastic control problem for a general one-dimensional diffusion, where time-inconsistency arises from a non-exponential discount function. To address this, we adopt a game-theoretic framework and study the optimality of a novel class of controls that encompasses both traditional singular controls -- responsible for generating multiple jumps and reflective boundaries (strong thresholds) -- and new mild threshold control strategies, which allow for the explosion of the control rate in absolutely continuous controls, thereby creating an inaccessible boundary (mild threshold) for the controlled process. We establish a general verification theorem, formulated in terms of a system of variational inequalities, that provides both necessary and sufficient conditions for equilibrium within the proposed class of control strategies and their combinations. To demonstrate the applicability of our theoretical results, we examine case studies in inventory management. We show that for certain parameter values, the problem admits a strong threshold control equilibrium in the form of Skorokhod reflection. In contrast, for other parameter values, we prove that no such equilibrium exists, necessitating the use of our extended control class. In the latter case, we explicitly construct an equilibrium using a mild threshold control strategy with a discontinuous, increasing, and exploding rate that induces an inaccessible boundary for the optimally controlled process, marking the first example of a singular control problem with such a solution structure.