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The Riesz representation theorem and weak\(^\ast\) compactness of semimartingales. (English) Zbl 1453.60098

Summary: We show that the sequential closure of a family of probability measures on the canonical space of càdlàg paths satisfying Stricker’s uniform tightness condition is a weak\(^\ast\) compact set of semimartingale measures in the dual pairing of bounded continuous functions and Radon measures, that is, the dual pairing from 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 characterisation of the strongest topology on the Skorokhod space for which these results are true.

MSC:

60G44 Martingales with continuous parameter
28C05 Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures
54D30 Compactness
60B05 Probability measures on topological spaces
60G05 Foundations of stochastic processes
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