Equilibria for Time-inconsistent Regular-singular Control Problems
Abstract
This paper studies a time-inconsistent mixed regular-singular control problem in continuous time with non-exponential discount. We seek time-consistent equilibria in an intrapersonal game framework and propose a novel definition of regular-singular equilibrium where both regular and singular controls are unified via control laws. We establish a verification theorem to provide a sufficient condition of the equilibrium. Furthermore, we propose a novel concept of mild equilibrium for which perturbations are applied separately to the regular and singular control and give the mild verification theorem. We illustrate the applicability of the theory to the effort and dividend problem and obtain an explicit equilibrium depending on the effort and dividend thresholds under a mixture of exponential discount functions and a pseudo-exponential discount function by solving the extended Hamilton--Jacobi--Bellman (HJB) system. The convexity of the value function is rigorously established, and the equilibrium conditions are verified. In addition, we establish the existence of the effort and dividend thresholds under a mixture of exponential discount functions. Numerical results reveal the impacts of exogenous parameters on the effort and dividend thresholds and the equilibrium value function, along with their economic implications.
Related
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- Entropy on regular sets and periodic-orbit growth for singular flows
BibTeX
@misc{jia2026equilibria,
title = {{Equilibria for Time-inconsistent Regular-singular Control Problems}},
author = {Yuting Jia and Zongxia Liang and Xiaodong Luo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08877},
eprint = {2609.08877},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08877},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Equilibria for Time-inconsistent Regular-singular Control Problems
AU - Yuting Jia
AU - Zongxia Liang
AU - Xiaodong Luo
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08877
UR - https://arxiv.org/abs/2609.08877
AB - This paper studies a time-inconsistent mixed regular-singular control problem in continuous time with non-exponential discount. We seek time-consistent equilibria in an intrapersonal game framework and propose a novel definition of regular-singular equilibrium where both regular and singular controls are unified via control laws. We establish a verification theorem to provide a sufficient condition of the equilibrium. Furthermore, we propose a novel concept of mild equilibrium for which perturbations are applied separately to the regular and singular control and give the mild verification theorem. We illustrate the applicability of the theory to the effort and dividend problem and obtain an explicit equilibrium depending on the effort and dividend thresholds under a mixture of exponential discount functions and a pseudo-exponential discount function by solving the extended Hamilton--Jacobi--Bellman (HJB) system. The convexity of the value function is rigorously established, and the equilibrium conditions are verified. In addition, we establish the existence of the effort and dividend thresholds under a mixture of exponential discount functions. Numerical results reveal the impacts of exogenous parameters on the effort and dividend thresholds and the equilibrium value function, along with their economic implications.
ER -