A Measure-Valued Obstacle Problem for an Obliquely Reflected Diffusion with a Max-Type Payoff

We study an obliquely reflected optimal stopping problem in the nonnegative quadrant with nonsmooth max-type payoff $G(x)=x_1\vee \alpha x_2$, and we develop a measure-valued potential-theoretic formulation of the associated obstacle problem. The kink of $G$ on the diagonal $x_1=\alpha x_2$ produces a singular surface measure in the distributional generator, while the oblique reflection directions generate boundary local-time contributions on the coordinate faces. Together with the absolutely continuous stopping gain, these terms define a total signed stopping measure $\text{Gtot}$. We derive the corresponding reflected Itô–Tanaka identity, prove a killed-resolvent representation of the value function in the continuation region, and show that the unrestricted reflected resolvent is generally incorrect because the process is not absorbed on the stopping set. The free boundary is formulated through a continuation-side trace condition for the killed potential. Under a vertical monotonicity hypothesis on $V-G$, the stopping set is shown to have an epigraph form. We finally prove a verification theorem: any admissible epigraph candidate satisfying contact, strict continuation, reflected Neumann compatibility, growth, the trace condition, and measure-superharmonicity coincides with the value function, and its first entry time is optimal.