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

Abstract

We show that the sequential closure of a family of probability measures on the canonical space of c{à}dl{à}g paths satisfying Stricker's uniform tightness condition is a weak${}^{\ast }$ 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.

V1-V6 differ substantially from v7-v8 in exposition v9 adds lemma 3.5, example 3.9, remark 5.6, proposition 5.7 (i) and some other minor remarks