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 \(G(x_1,x_2)=x_1\veeαx_2\). The main novelty is a measure-valued variational formulation: the stopping gain \(Γ=c+rG-\mathcal LG\) 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 \[ V=G-R_r^{\mathcal C}Γ, \] 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.