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

S. S. Gabriyelyan ()
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S. S. Gabriyelyan: Ben-Gurion University of the Negev

Journal of Theoretical Probability, 2011, vol. 24, issue 3, 595-614

Abstract: Abstract Let μ and ν be fixed probability measures on a filtered space $(\varOmega, \mathcal{F}, \allowbreak(\mathcal{F}_{t} )_{t\in \mathbf{R}^{+} } )$ . Denote by μ T and ν T (respectively, μ T− and ν T−) the restrictions of the measures μ and ν on $\mathcal{F}_{T} $ (respectively, on $\mathcal{F}_{T-} $ ) for a stopping time T. We find the Hahn decomposition of μ T and ν T using the Hahn decomposition of the measures μ, ν and the Hellinger process h t in the strict sense of order $\frac{1}{2}$ . The norm of the absolutely continuous component of μ T− with respect to ν T− is computed in terms of density processes and Hellinger integrals.

Keywords: Density processes; Hellinger integrals; Hellinger processes; The Hahn decomposition; Stopping times; Absolute continuity and singularity; 60G07 (search for similar items in EconPapers)
Date: 2011
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DOI: 10.1007/s10959-011-0359-2

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