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

S. S. Gabriyelyan

doi:10.1007/s10959-011-0359-2

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

S. S. Gabriyelyan

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Let μ and ν be fixed probability measures on a filtered space (Ω, F, (F_t)_t∈R+). Denote by μT and νT (respectively, μT- and νT-) the restrictions of the measures μ and ν on F_T (respectively, on F_T-) 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 h_t 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 language	English
Pages (from-to)	595-614
Number of pages	20
Journal	Journal of Theoretical Probability
Volume	24
Issue number	3
DOIs	https://doi.org/10.1007/s10959-011-0359-2
State	Published - 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
General Mathematics
Statistics, Probability and Uncertainty

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10.1007/s10959-011-0359-2

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title = "Absolute Continuity and Singularity of Two Probability Measures on a Filtered Space",

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.",

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N2 - 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.

AB - 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.

KW - Absolute continuity and singularity

KW - Density processes

KW - Hellinger integrals

KW - Hellinger processes

KW - Stopping times

KW - The Hahn decomposition

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DO - 10.1007/s10959-011-0359-2

M3 - Article

AN - SCOPUS:79960846645

SN - 0894-9840

VL - 24

SP - 595

EP - 614

JO - Journal of Theoretical Probability

JF - Journal of Theoretical Probability

IS - 3

ER -

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

Abstract

Keywords

ASJC Scopus subject areas

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