A canonical setting and separating times for continuous local martingales
type: Item
instance of: scholarly article
title: A canonical setting and separating times for continuous local martingales
zbMATH Open document ID: 1178.60033
author: J. Martínez
published in: Stochastic Processes and their Applications
publication date: 2009-05-06T00:00:00Z
review text: The authors construct a certain canonical setting for continuous local martingales (CLMs) and find an explicit form of separating times [see \textit{A. S. Cherny} and \textit{M. A. Urusov}, Theory Probab. Appl. 48, No. 2, 337--347 (2003; Zbl 1070.60004), From stochastic calculus to mathematical finance. The Shiryaev Festschrift. Berlin: Springer. 125--168 (2006; Zbl 1103.60043)] for CLMs in this setting. Informally, a separating time for two measures \(\mathsf P\) and \(\widetilde{\mathsf P}\) on a filtered space \((\Omega,\mathcal F,(\mathcal F_t)_{t\geq0})\) is a stopping time \(S\) such that \(\mathsf P\) and \(\widetilde{\mathsf P}\) are equivalent before \(S\) and singular after it. Separating times can be used to establish criteria for absolute continuity and singularity between measures. The idea of the canonical setting for CLMs relies on the Dambis--Dubins--Schwarz theorem, which states that each CLM \(X\) starting from~0 is a time-changed Brownian motion. As canonical space for CLMs the authors consider the space \(\Omega^*\) of pairs \((\beta,\alpha)\) of a Brownian trajectory and a time-change trajectory. They prove that there is a one-to-one correspondence between distributions of CLMs and probability kernels from the Wiener space to the time-change path space; these kernels are non-anticipating in a certain sense. Given a non-anticipating kernel \(K\) one can define a probability measure \(\mathsf P\) on \(\Omega^*\) such that under \(\mathsf P\) the process \(\beta_{\alpha_.}\) is a ``canonical version'' on \(\Omega^*\) of CLMs corresponding to \(K\). The authors further discuss pure CLMs (PCLMs). They present several new definitions of PCLMs \(X\) (covering also the case \(\mathsf P(\langle X\rangle_\infty<\infty)>0\)), and show they are equivalent to definitions already existing in the literature. The separating time for measures \(\mathsf P\) and \(\widetilde{\mathsf P}\) on \(\Omega^*\) corresponding to non-anticipating kernels \(K\) and \(\wideti…
reviewed by: Antonis Papapantoleon
Mathematics Subject Classification ID: 60G44
Mathematics Subject Classification ID: 60G30
Mathematics Subject Classification ID: 60G48
Mathematics Subject Classification ID: 60G40
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60H10
zbMATH DE Number: 5551371
zbMATH Keywords: separating times
zbMATH Keywords: continuous local martingales
zbMATH Keywords: Brownian motion
zbMATH Keywords: canonical setting
zbMATH Keywords: Dambis--Dubins--Schwarz theorem
zbMATH Keywords: pure continuous local martingales
MaRDI profile type: Publication
full work available at URL: j.spa.2008.05.003
OpenAlex ID: W2003792001
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DOI: 10.1016/J.SPA.2008.05.003
description: scientific article; zbMATH DE number 5551371