Finitely Additive Supermartingales

TL;DR

This paper revisits finitely additive supermartingales, exploring their properties and applications in measure decomposition and stochastic process theory, including versions of classical theorems without relying on exogenous probability measures.

Contribution

It develops the theory of finitely additive supermartingales, providing new measure decomposition results and versions of the Doob Meyer and Bichteler-Dellacherie theorems without external probability measures.

Findings

Established a version of the Doob Meyer decomposition for finitely additive supermartingales.

Proved a version of the Bichteler and Dellacherie theorem without exogenous probability measures.

Enhanced understanding of measure decompositions over filtered probability spaces.

Abstract

The concept of finitely additive supermartingales, originally due to Bochner, is revived and developed. We exploit it to study measure decompositions over filtered probability spaces and the properties of the associated Dol\'{e}ans-Dade measure. We obtain versions of the Doob Meyer decomposition and, as an application, we establish a version of the Bichteler and Dellacherie theorem with no exogenous probability measure.