Nonsmooth Obstacles and Killed Resolvents in Reflected Stochastic Control

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 $\mathrm{\Gamma }=c+rG-\mathcal{L}G$ 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 $$V=G-R_r^{\mathcal{C}}\mathrm{\Gamma },$$ 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.