Nonsmooth Obstacles and Killed Resolvents in Reflected Stochastic Control

Louis Shuo Wang; Jiguang Yu

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arXiv · 2606.21575

Nonsmooth Obstacles and Killed Resolvents in Reflected Stochastic Control

Louis Shuo Wang·Jiguang Yu

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Abstract

We study an infinite-horizon optimal stopping problem for a normally reflected two-dimensional diffusion in the quadrant with nonsmooth max-type payoff $G(x_1,x_2)=x_1\vee\alpha x_2$. The main novelty is a measure-valued variational formulation: the stopping gain $\Gamma=c+rG-\mathcal LG$ is shown to be a signed Radon measure whose singular component is supported on the kink diagonal $\{x_1=\alpha x_2\}$, and this component is computed explicitly. We prove that the value admits the killed-resolvent representation \[ V=G-R_r^{\mathcal C}\Gamma, \] 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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Louis Shuo Wang, Jiguang Yu. 2026-06-19. Nonsmooth Obstacles and Killed Resolvents in Reflected Stochastic Control. https://arxiv.org/abs/2606.21575

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