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.
From: Louis Shuo Wang [view email]
[v1]
Fri, 19 Jun 2026 16:18:17 UTC (129 KB)
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