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Is there a predictable criterion for mutual singularity of two probability measures on a filtered space?

Schachermayer, Walter and Schachinger, Werner (1999) Is there a predictable criterion for mutual singularity of two probability measures on a filtered space? Working Papers SFB "Adaptive Information Systems and Modelling in Economics and Management Science", 26. SFB Adaptive Information Systems and Modelling in Economics and Management Science, WU Vienna University of Economics and Business, Vienna.

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Abstract

The theme of providing predictable criteria for absolute continuity and for mutual singularity of two density processes on a filtered probability space is extensively studied, e.g., in the monograph by J. Jacod and A. N. Shiryaev [JS]. While the issue of absolute continuity is settled there in full generality, for the issue of mutual singularity one technical difficulty remained open ([JS], p210): "We do not know whether it is possible to derive a predictable criterion (necessary and sufficient condition) for "P'T..." (expression not representable in this abstract). It turns out that to this question raised in [JS] which we also chose as the title of this note, there are two answers: on the negative side we give an easy example, showing that in general the answer is no, even when we use a rather wide interpretation of the concept of "predictable criterion". The difficulty comes from the fact that the density process of a probability measure P with respect to another measure P' may suddenly jump to zero. On the positive side we can characterize the set, where P' becomes singular with respect to P - provided this does not happen in a sudden but rather in a continuous way - as the set where the Hellinger process diverges, which certainly is a "predictable criterion". This theorem extends results in the book of J. Jacod and A. N. Shiryaev [JS]. (author's abstract)

Item Type: Paper
Keywords: continuity and singularity of probability measures / Hellinger processes / stochastic integrals / stopping times
Divisions: Departments > Finance, Accounting and Statistics > Statistics and Mathematics
Departments > Informationsverarbeitung u Prozessmanag. > Informationswirtschaft
Departments > Informationsverarbeitung u Prozessmanag. > Produktionsmanagement > Taudes
Departments > Marketing > Service Marketing und Tourismus
Depositing User: Repository Administrator
Date Deposited: 28 Feb 2002 15:18
Last Modified: 26 Jan 2011 13:06
URI: http://epub.wu.ac.at/id/eprint/1600

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