Supermartingales as Radon-Nikodym densities and related measure extensions
type: Item
instance of: scholarly article
title: Supermartingales as Radon-Nikodym densities and related measure extensions
zbMATH Open document ID: 1356.60070
DOI: 10.1214/14-AOP956
author: Nicolas Perkowski
author: Johannes Ruf
published in: The Annals of Probability
publication date: 2016-02-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/1309.4623
full work available at URL: https://projecteuclid.org/euclid.aop/1449843627
review text: The authors compare two approaches of associating countably and finitely additive measures to supermartingales. In the countably additive case, they prove the existence of a probability measure such that a given supermartingale can be interpreted as Radon-Nikodym density of that measure. In particular, it is shown that this probability measure can be constructed on the canonical space itself and not only on the product space, even if the given supermartingale is not a local martingale. Moreover, precise necessary and sufficient conditions are provided for the uniqueness of such a probability measure associated to the given supermartingale. In the finitely additive case, the existence of a finitely additive measure associated to the given semimartingale is shown as long as the filtering probability space is sufficiently rich, supporting a Brownian motion. It is also shown that such a finitely additive measure is never unique. The argument for the existence of such finitely additive measures also yields the existence of uniformly integrable martingales that Fatou converge to a given supermartingale.
reviewed by: Pavel Gapeev
Mathematics Subject Classification ID: 60G48
Mathematics Subject Classification ID: 60G44
Mathematics Subject Classification ID: 60A10
Mathematics Subject Classification ID: 60H99
Mathematics Subject Classification ID: 60J65
zbMATH DE Number: 6541354
zbMATH Keywords: supermartingales
zbMATH Keywords: Radon-Nikodym densities
zbMATH Keywords: finitely additive measure
zbMATH Keywords: Föllmer measure
zbMATH Keywords: Fatou convergence
zbMATH Keywords: Carathéodory extension theorem
MaRDI profile type: Publication
arXiv ID: 1309.4623
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description: scientific article; zbMATH DE number 6541354