Nonsmooth Obstacles and Killed Resolvents in Reflected Stochastic Control
Abstract
We study an infinite-horizon optimal stopping problem for a normally reflected two-dimensional diffusion in the quadrant with nonsmooth max-type payoff G(x1,x2)=x1∨αx2. The main novelty is a measure-valued variational formulation: the stopping gain Γ=c+rG−ℒG is shown to be a signed Radon measure whose singular component is supported on the kink diagonal {x1=αx2}, and this component is computed explicitly. We prove that the value admits the killed-resolvent representation V=G−Rr𝒞Γ, where the reflected diffusion is killed upon entry into the stopping set. This corrects the generally invalid unrestricted-resolvent formula. Under explicit monotonicity hypotheses, the stopping set has epigraph form, and the free boundary is characterized by a killed-potential trace condition. A verification theorem certifies locally Lipschitz candidate boundaries as optimal.
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BibTeX
@misc{wang2026nonsmooth,
title = {{Nonsmooth Obstacles and Killed Resolvents in Reflected Stochastic Control}},
author = {Louis Shuo Wang and Jiguang Yu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.21575},
eprint = {2606.21575},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.21575},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Nonsmooth Obstacles and Killed Resolvents in Reflected Stochastic Control
AU - Louis Shuo Wang
AU - Jiguang Yu
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2606.21575
UR - https://arxiv.org/abs/2606.21575
AB - We study an infinite-horizon optimal stopping problem for a normally reflected two-dimensional diffusion in the quadrant with nonsmooth max-type payoff G(x_1,x_2)=x_1∨αx_2. The main novelty is a measure-valued variational formulation: the stopping gain Γ=c+rG−ℒG is shown to be a signed Radon measure whose singular component is supported on the kink diagonal {x_1=αx_2}, and this component is computed explicitly. We prove that the value admits the killed-resolvent representation V=G−R_r^𝒞Γ, where the reflected diffusion is killed upon entry into the stopping set. This corrects the generally invalid unrestricted-resolvent formula. Under explicit monotonicity hypotheses, the stopping set has epigraph form, and the free boundary is characterized by a killed-potential trace condition. A verification theorem certifies locally Lipschitz candidate boundaries as optimal.
ER -