The Finite-Horizon Retirement Problem with Borrowing Constraint: A Zero-Sum Stopper Versus Singular-Controller Game

This paper investigates an optimal consumption, investment, and early retirement problem in the presence of a mandatory retirement date and a borrowing constraint that prohibits borrowing against future labor income during employment. To handle the borrowing constraint, we employ a dual-martingale approach and reformulate the problem as a finite-horizon, two-player, zero-sum game between a singular controller and a stopper. The value of the game is characterized by a parabolic variational inequality with both obstacle and gradient constraints, giving rise to two time-dependent free boundaries that determine the optimal retirement threshold and the wealth-binding region. Using advanced PDE techniques and nonstandard analytical arguments, we prove the existence and uniqueness of a strong solution to the variational inequality and derive key properties of the free boundaries, including their monotonicity and smoothness. We further establish that the solution coincides with the game value and provide a duality theorem that characterizes the optimal strategy. To the best of our knowledge, this is the first study in the mathematical finance literature to analyze a finite-horizon, zero-sum game involving both singular control and stopping, offering novel insights into the joint dynamics of retirement timing, consumption, and portfolio choice.

Funding: T. Kim is supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) [Grant RS-2024-00351151]. Z. Yang is supported by the National Natural Science Foundation of China [Grants 12571512 and 12371470] and the Guangdong Basic and Applied Basic Research Foundation [Grant 2026A1515011188]. J. Jeon is supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) [Grant RS-2023-00212648].

Supplemental Material: The online appendix is available at https://doi.org/10.1287/moor.2025.0873.