Absolute continuity and singularity of two probability measures on a filtered space

2008/02/04 by Saak Gabriyelyan, S. S. Gabriyelyan, Gabriyelyan, S. S.
Economics, Econometrics and Finance · Mathematics · #60G07 #60G99 #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.PR #msc:60G07 #msc:60G99

paper · pdf · doi:10.48550/arxiv.0802.0385

18 pages, no figures

openalex publication_date 2008/02/04 · arxiv created 2011/04/06 · arxiv updated 2011/04/07 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28

Abstract

Let μ and ν be fixed probability measures on a filtered space (Ω, \cal F, (\cal F_t)_t∈ \bf R⁺). Denote by μ_T and ν_T (respectively, μ_T- and ν_T- ) the restrictions of the measures μ and ν on \cal F_T (respectively, on \cal 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 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.

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