Abstract
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∗ 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.
Highlights
The Riesz representation theorem states that the operation of integration defines a one-to-one correspondence between the continuous linear functionals on the bounded continuous functions and the Radon measures on a topological space
We relate Stricker’s uniform tightness condition of semimartingales to the weak∗ compactness in the dual pairing of bounded continuous functions and Radon measures, that is, the dual pairing from the Riesz representation theorem on the canonical space of càdlàg paths
We address the problem of regularity by introducing a new weak topology on the Skorokhod space that has the same continuous functions as the S-topology, suitable compact sets and satisfies a strong separation axiom
Summary
The Riesz representation theorem states that the operation of integration defines a one-to-one correspondence between the continuous linear functionals on the bounded continuous functions and the Radon measures on a topological space. We relate Stricker’s uniform tightness condition of semimartingales to the weak∗ compactness in the dual pairing of bounded continuous functions and Radon measures, that is, the dual pairing from the Riesz representation theorem on the canonical space of càdlàg paths. The objective of the paper is to provide a weak∗ compactness result for càdlàg semimartingales under the most general topological assumption on the path space. Our main contribution is to unify the previous results on the weak convergence of semimartingales and provide an easy method for constructing weak∗ compact sets of semimartingales on the canonical space of càdlàg paths. By Mt (D) we denote the family of Radon measures (of finite total variation) on the Skorokhod space D, while Mτ (D) P(Rd )) we denote the family of all probability measures on the Skorokhod space D All discussion on these properties generalises as such for relative topologies on closed sets
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