Minimal Reflected BSDEs and Superhedging of Irregular American Claims

We study the superhedging of American claims with an irregular optional payoff process $ξ$ over the full set M(X) of equivalent local martingale measures of a locally bounded market. We show that a process which is a strong optional supermartingale of class (D) under every Q(X), with no bound from either side, decomposes as S=S₀+∫ H\,dX-A-C₋, with a common integrand H, a right-continuous increasing process A and a purely discontinuous increasing process C given explicitly by the right jumps of S. For the robust Snell envelope this yields the superhedging duality p₀(ξ)=Q_τEQ[ξ_τ] with an explicit optimal strategy and no regularity of $ξ$ in time. Unbounded payoffs are covered when \ξ_τ\ is uniformly integrable jointly over measures and stopping times, or when $|ξ|$ is dominated by a traded uniformly integrable martingale. For payoffs bounded below, the price is attained by wealth bounded below by a constant. For payoffs unbounded below this can fail, already in a complete market, and the price is attained once the constant bound is replaced by $-M$, with M a traded martingale dominating $|ξ|$. In an incomplete Brownian market we study the associated reflected BSDE with an optional obstacle and a Lipschitz driver depending on the value. We prove existence and uniqueness of the minimal supersolution, a representation at stopping times, comparison and an equivalent dynamic minimality condition, for bounded data and for data dominated by a traded martingale. For a European-type obstacle, the Skorokhod conditions on the total regulator hold for some supersolution if and only if the terminal claim is attainable by a bounded wealth process; so in an incomplete market some data admit no supersolution satisfying them, while the right-jump regulator always satisfies its contact condition.