Quasi-sure analysis, aggregation and dual representations of sublinear expectations in general spaces

TL;DR

This paper develops a framework for analyzing sublinear expectations without a dominating measure, providing new representations, aggregation properties, and conditions for consistency in a quasi-sure setting.

Contribution

It introduces a novel quasi-sure approach to conditional expectations and sublinear expectations in general spaces, extending classical theory without requiring a dominating measure.

Findings

Constructs linear conditional expectations in a quasi-sure sense.

Establishes an aggregation property for sublinear expectations.

Provides an equivalence between consistency and a pasting property of measures.

Abstract

We consider coherent sublinear expectations on a measurable space, without assuming the existence of a dominating probability measure. By considering a decomposition of the space in terms of the supports of the measures representing our sublinear expectation, we give a simple construction, in a quasi-sure sense, of the (linear) conditional expectations, and hence give a representation for the conditional sublinear expectation. We also show an aggregation property holds, and give an equivalence between consistency and a pasting property of measures.