Minimal Reflected BSDEs and Superhedging of Irregular American Claims

@misc{bouhadou2026minimal,
  title = {{Minimal Reflected BSDEs and Superhedging of Irregular American Claims}},
  author = {Siham Bouhadou and Imade Fakhouri and Youssef Ouknine},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.05116},
  eprint = {2611.05116},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.05116},
}

TY  - UNPB
TI  - Minimal Reflected BSDEs and Superhedging of Irregular American Claims
AU  - Siham Bouhadou
AU  - Imade Fakhouri
AU  - Youssef Ouknine
PY  - 2026
DA  - 2026/11/03
PB  - arXiv
DO  - 10.48550/arxiv.2611.05116
UR  - https://arxiv.org/abs/2611.05116
AB  - 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∈ℳ(X), with no bound from either side, decomposes as S=S_0+∫ 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_Qsup_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.
ER  -