Absolute Continuity and Singularity of Two Probability Measures on a Filtered Space

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Abstract

Let μ and ν be fixed probability measures on a filtered space (Ω, F, (Ft)t∈R+). Denote by μT and νT (respectively, μT- and νT-) the restrictions of the measures μ and ν on FT (respectively, on FT-) for a stopping time T. We find the Hahn decomposition of μT and νT using the Hahn decomposition of the measures μ, ν and the Hellinger process ht in the strict sense of order 1/2. The norm of the absolutely continuous component of μT- with respect to νT- is computed in terms of density processes and Hellinger integrals.

Original languageEnglish
Pages (from-to)595-614
Number of pages20
JournalJournal of Theoretical Probability
Volume24
Issue number3
DOIs
StatePublished - 1 Sep 2011

Keywords

  • Absolute continuity and singularity
  • Density processes
  • Hellinger integrals
  • Hellinger processes
  • Stopping times
  • The Hahn decomposition

ASJC Scopus subject areas

  • Statistics and Probability
  • Mathematics (all)
  • Statistics, Probability and Uncertainty

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