Superreplication under Model Uncertainty in Discrete Time

We study the superreplication of contingent claims under model uncertainty in discrete time. We show that optimal superreplicating strategies exist in a general measure-theoretic setting; moreover, we characterize the minimal superreplication price as the supremum over all continuous linear pricing functionals on a suitable Banach space. The main ingredient is a closedness result for the set of claims which can be superreplicated from zero capital; its proof relies on medial limits.Comment: 14 pages; forthcoming in 'Finance and Stochastics

Precompact noncompact reflexive abelian groups

We present a series of examples of precompact, noncompact, reflexive topological Abelian groups. Some of them are pseudocompact or even countably compact, but we show that there exist precompact non-pseudocompact reflexive groups as well. It is also proved that every pseudocompact Abelian group is a quotient of a reflexive pseudocompact group with respect to a closed reflexive pseudocompact subgroup

A generalization of Markov's approach to the continuity problem for Type 1 computable functions

We axiomatize and generalize Markov's approach to the continuity problem for Type 1 computable functions, i.e. the problem of finding sufficient conditions on a computable topological space to obtain a theorem of the form "computable functions are (effectively) continuous". In a computable topological space, a point $x$ is called effectively adherent to a set $A$ if there is an algorithm that on input a neighborhood of $x$ produces a point of $A$ in that neighborhood. We say that a space $X$ satisfies a Markov condition if, whenever a point $x$ of $X$ is effectively adherent to a subset $A$ of $X$, the singleton $\{x\}$ is not a semi-decidable subset of $A\cup\{x\}$. We show that this condition prevents functions whose domain is $X$ from having effective discontinuities, provided that their codomain is a space where points have neighborhood bases of co-semi-decidable sets. We then show that results that forbid effective discontinuities can be turned into (abstract) continuity results on spaces where the closure and effective closure of semi-decidable sets naturally agree -this happens for instance on spaces which admit a dense and computable sequence. This work is motivated by the study of the space of marked groups, for which the author has shown that most known continuity results (of Ceitin, Moschovakis, Spreen) do not apply.Comment: 21 page