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Teoriya Veroyatnostei i ee Primeneniya, 1999, Volume 44, Issue 1, Pages 101–110
DOI: https://doi.org/10.4213/tvp600
(Mi tvp600)
 

This article is cited in 4 scientific papers (total in 4 papers)

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

W. Schachermayera, W. Schachingerb

a Department of Statistics, University of Vienna, Austria
b Financial and Actuarial Mathematics Group, Technical University of Vienna, Austria
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 [Limit Theorems for Stochastic Processes, Springer-Verlag, Berlin, 1987]. While the issue of absolute continuity is settled there in full generality, for the issue of mutual singularity one technical difficulty remained open [J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, Springer-Verlag, Berlin, 1987, p. 210]: "We do not know whether it is possible to derive a predictable criterion (necessary and sufficient condition) for $P_T'\perp P_T,\ldots$". It turns out that there are two answers to this question raised in the monograph of J. Jacod and A. N. Shiryaev: 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 happens in a continuous way rather than suddenly—as the set where the Hellinger process diverges, which certainly is a "predictable criterion." This theorem extends results in the monograph of J. Jacod and A. N. Shiryaev.
Keywords: continuity and singularity of probability measures, Hellinger processes, stochastic integrals, stopping times.
Received: 06.03.1998
English version:
Theory of Probability and its Applications, 2000, Volume 44, Issue 1, Pages 51–59
DOI: https://doi.org/10.1137/S0040585X97977367
Bibliographic databases:
Language: English
Citation: W. Schachermayer, W. Schachinger, “Is there a predictable criterion for mutual singularity of two probability measures on a filtered space?”, Teor. Veroyatnost. i Primenen., 44:1 (1999), 101–110; Theory Probab. Appl., 44:1 (2000), 51–59
Citation in format AMSBIB
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\by W.~Schachermayer, W.~Schachinger
\paper Is there a predictable criterion for mutual singularity of two probability measures on a~filtered space?
\jour Teor. Veroyatnost. i Primenen.
\yr 1999
\vol 44
\issue 1
\pages 101--110
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\crossref{https://doi.org/10.4213/tvp600}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1751191}
\zmath{https://zbmath.org/?q=an:0963.60036}
\transl
\jour Theory Probab. Appl.
\yr 2000
\vol 44
\issue 1
\pages 51--59
\crossref{https://doi.org/10.1137/S0040585X97977367}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000087555000005}
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  • https://www.mathnet.ru/eng/tvp/v44/i1/p101
  • This publication is cited in the following 4 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Теория вероятностей и ее применения Theory of Probability and its Applications
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