Reflected Optimal Stopping with a Max-Type Payoff: Measure-Valued Stopping Gains and Killed Resolvent Representation

We study an infinite-horizon optimal stopping problem for a two-dimensional normally reflected diffusion in the quadrant with payoff $G(x_1,x_2)=x_1\vee \alpha x_2$. The problem combines three features that complicate the usual free-boundary analysis: reflection on the coordinate axes, a genuinely two-dimensional stopping region, and a nonsmooth max-type reward. We formulate the associated reflected obstacle problem, prove a verification theorem under explicit Itô–Krylov–Tanaka admissibility and measure-superharmonicity assumptions, and derive a conditional epigraph structure for the stopping set. The main technical point is that the stopping-gain object $\mathrm{\Gamma }=c+rG-\mathcal{L}G$ is a signed measure rather than a function. Its diagonal component is ${\mathrm{\Gamma }}^{\mathrm{\Delta }}(dx)=-\frac{n^{\mathrm{\top }}a(x)n}{2\sqrt{1+{\alpha }^2}}{\sigma }_{\mathrm{\Delta }}(dx)$, $n=(1,-\alpha )$, which shows that pointwise stopping-gain sign conditions must be interpreted with care. We also prove that the correct potential representation is the killed-resolvent formula $V(x)=G(x)-R_r^{\mathcal{C}}\mathrm{\Gamma }(x)$, rather than the unrestricted reflected resolvent. A constant-coefficient reflected Brownian example illustrates the diagonal singular term explicitly.