Natural processes and Doob-Meyer decompositions over a probability gage space
type: Item
instance of: scholarly article
title: Natural processes and Doob-Meyer decompositions over a probability gage space
zbMATH Open document ID: 0552.46032
DOI: 10.1016/0022-1236(84)90046-6
published in: Journal of Functional Analysis
publication date: 1984-01-01T00:00:00Z
review text: The paper is devoted to a non-commutative generalization of natural processes in the sense of Meyer. In Section 1 the naturalness is reformulated so that it may be used in the non-commutative setting. In Section 2 probability gage spaces (\({\mathcal U},\tau)\) are considered where \({\mathcal U}\) is a finite von Neumann algebra, \(\tau\) is a faithful normal probability trace on \({\mathcal U}\). In this setting natural processes are introduced and some examples are given. This enables the authors to prove that Doob-Meyer decomposition obtained earlier by them and \textit{R. F. Streater} [ibid. 48, 172-212 (1982; Zbl 0492.46051)] is unique. Here they also introduce Nelson's ''regular processes'' in this non-commutative setting and show that \((| X_ t|^ 2)\) is a regular process for a wide class of martingales \((X_ t)\). In Section 3 the results of the previous sections are applied in the case of the Clifford probability gage spaces.
Mathematics Subject Classification ID: 46L51
Mathematics Subject Classification ID: 46L53
Mathematics Subject Classification ID: 46L54
Mathematics Subject Classification ID: 46L55
Mathematics Subject Classification ID: 46L60
Mathematics Subject Classification ID: 60G48
zbMATH DE Number: 3879543
zbMATH Keywords: non-commutative generalization of natural processes
zbMATH Keywords: faithful normal probability trace
zbMATH Keywords: Doob-Meyer decomposition
zbMATH Keywords: regular processes
zbMATH Keywords: Clifford probability gage spaces
author: Christopher Barnett
author: Ivan F. Wilde
MaRDI profile type: Publication
full work available at URL: 0022-1236(84)90046-6
OpenAlex ID: W2066221280
cites work: Supermartingales on Semifinite Von Neumann Algebras
cites work: The Ito-Clifford integral
cites work: Q3333817
cites work: Quasi-free quantum stochastic integrals for the CAR and CCR
cites work: Stochastic integrals in an arbitrary probability gauge space
cites work: A general bilinear vector integral
cites work: Q5512462
cites work: Q5552132
cites work: Correction to ''A non-commutative extension of abstract integration''
cites work: Non-commutative LP-spaces
description: scientific article; zbMATH DE number 3879543