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arXiv · 2610.06997

Local $W^{2,1}_{p}$ convergence of penalization for time-inconsistent stopping problems

Xiaodong Luo·Xiang Yu
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Abstract

This paper studies the penalization method for a general class of time-inconsistent stopping problems. Inspired by some existing results on penalization for optimal stopping problems, we introduce the penalized PDE problem and the penalized control problem for time-inconsistent stopping problems, uncovering some interesting discrepancy stemming from the time-inconsistency. As the penalization tends to infinity, we develop some novel arguments to rigorously establish the local $W^{2,1}_{p}$ convergence results for the solution of the penalized PDE problem to a solution of the variational inequality associated to the time-inconsistent stopping problem. Under an additional concavity assumption, we further derive the convergence of the penalized time-inconsistent control problem to the time-inconsistent stopping problem. We revisit two examples of time-inconsistent stopping to numerically illustrate the convergence theory.

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Xiaodong Luo, Xiang Yu. 2026-10-04. Local $W^{2,1}_{p}$ convergence of penalization for time-inconsistent stopping problems. https://arxiv.org/abs/2610.06997

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