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Local W2,1p 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 W2,1p 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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Cite

BibTeXRIS
BibTeX
@misc{luo2026local,
  title = {{Local $W^2,1_p$ convergence of penalization for time-inconsistent stopping problems}},
  author = {Xiaodong Luo and Xiang Yu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.06997},
  eprint = {2610.06997},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.06997},
}
RIS
TY  - UNPB
TI  - Local W^2,1_p convergence of penalization for time-inconsistent stopping problems
AU  - Xiaodong Luo
AU  - Xiang Yu
PY  - 2026
DA  - 2026/10/04
PB  - arXiv
DO  - 10.48550/arxiv.2610.06997
UR  - https://arxiv.org/abs/2610.06997
AB  - 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.
ER  -