The Riesz representation theorem and weak${}^*$ compactness of semimartingales

TL;DR

This paper establishes the weak* compactness of families of semimartingale measures under tightness conditions, using the Riesz representation theorem, and characterizes the relevant topologies on the Skorokhod space.

Contribution

It provides a novel application of the Riesz representation theorem to characterize the weak* compactness of semimartingale measures on path spaces.

Findings

Sequential closure of probability measures is weak* compact.

Results extend to quasi- and supermartingales.

Characterization of the strongest topology on Skorokhod space.

Abstract

We show that the sequential closure of a family of probability measures on the canonical space of c{\`a}dl{\`a}g paths satisfying Stricker's uniform tightness condition is a weak${}^*$ compact set of semimartingale measures in the pairing of the Riesz representation theorem under topological assumptions on the path space. Similar results are obtained for quasi- and supermartingales under analogous conditions. In particular, we give a full characterization of the strongest topology on the Skorokhod space for which these results are true.