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

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  • S. S. Gabriyelyan

    (Ben-Gurion University of the Negev)

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.

Suggested Citation

  • S. S. Gabriyelyan, 2011. "Absolute Continuity and Singularity of Two Probability Measures on a Filtered Space," Journal of Theoretical Probability, Springer, vol. 24(3), pages 595-614, September.
    • Handle: RePEc:spr:jotpro:v:24:y:2011:i:3:d:10.1007_s10959-011-0359-2
    DOI: 10.1007/s10959-011-0359-2
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    References listed on IDEAS

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    1. Darwich, A. R., 2001. "About the absolute continuity and orthogonality for two probability measures," Statistics & Probability Letters, Elsevier, vol. 52(1), pages 1-8, March.
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