Minimal Reflected BSDEs and Superhedging of Irregular American Claims

Abstract

We study the superhedging of American claims with an irregular optional payoff process $ξ$ over the full set $\mathcal{M}(X)$ 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 $Q\in\mathcal{M}(X)$, with no bound from either side, decomposes as $S=S_0+\int H\,dX-A-C_-$, 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 $p_0(ξ)=\sup_{Q}\sup_τE^Q[ξ_τ]$ 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.