Optional Stopping for Superhedging Supermartingales

Mathematics > Probability [Submitted on 16 Jun 2026] Title:Optional Stopping for Superhedging Supermartingales View PDF HTML (experimental)Abstract:Superhedging supermartingales, introduced by the authors in previous work, are non-probabilistic processes defined via subadditive outer integrals that carry a purely financial interpretation in terms of superhedging cost. Building on the Leinert-König theory of non-lattice integration, the present paper establishes several results that are classical in probability theory but whose non-probabilistic proofs require fundamentally new arguments: (i) a tower inequality for the conditional outer integral \overline{\sigma}_j applied at stopping times, reducing to equality when the integrand is conditionally integrable; (ii) three versions of Doob's optional stopping theorem, organised by the class of supermartingale and the range of the stopping times; and (iii) Dubins' upcrossing inequality in both finite- and infinite-time horizons. A key structural result, property (K)-a.e., identifies conditions under which the two superhedging operators \overline{\sigma}_j and \overline{I}_j coincide on non-negative functions, extending the scope of all preceding results to the positive operator \overline{I}_j. None of the proofs invoke classical measure-theoretic tools; in particular, (classical) integrability and measurability are not assumed. The analogues of classical stochastic results acquire a purely financial interpretation and, in this way, gain depth and generality by providing a context that is independent of any a priori probabilistic structure. References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.