Nonlinear optimal stopping and domineering claims

We study nonlinear optimal stopping and domineering claims in a Brownian--Poisson filtration, under a nonlinear evaluation induced by a Lipschitz BSDE driver satisfying a comparison condition. A nonlinear domineering claim is a strong Eᶠ-martingale dominating the reward process at every stopping time. We first establish the existence of such claims for every optional reward in S². For left-limited, right-upper semicontinuous rewards, we prove a local dual representation at every stopping time: the nonlinear optimal stopping value equals the essential infimum of the values of all domineering claims defined on the remaining stochastic interval. This infimum is attained by a claim constructed from the associated reflected BSDE. We further show that a global domineering claim attains the value at a prescribed stopping time if and only if the reflection processes have accumulated no increase up to that time, with the appropriate endpoint convention. When the reward is right-upper semicontinuous in Eᶠ-expectation and the family of contact times is nonempty, its essential infimum is optimal. Under strict comparison, we also establish a necessary contact condition for optimality. No convexity assumption on the driver is required.