Sublinear expectations for countable-state uncertain processes

TL;DR

This paper extends sublinear expectations for countable-state uncertain processes from bounded functions to a broader class of real measurable functions on cdlg paths, enhancing their applicability.

Contribution

It introduces a method to extend sublinear expectations to richer function spaces in the countable-state setting under mild conditions.

Findings

Extension to real measurable functions on cdlg paths achieved

Conditions under which sublinear Markov semigroups induce suitable expectations identified

Broader domain for sublinear expectations in countable-state processes established

Abstract

Sublinear expectations for uncertain processes have received a lot of attention recently, particularly methods to extend a downward-continuous sublinear expectation on the bounded finitary functions to one on the non-finitary functions. In most of the approaches the domain of the extension is not very rich because it is limited to bounded measurable functions on the set of all paths. This contribution alleviates this problem in the countable-state case by extending, under a mild condition, to the extended real measurable functions on the set of c\`adl\`ag paths, and investigates when a sublinear Markov semigroup induces a sublinear expectation that satisfies this mild condition.