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 $(Ω, {\cal F}, ({\cal F}_t)_{t\in {\bf R}^{+}})$. 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.