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

TL;DR

This paper investigates the conditions under which two probability measures on a filtered space are absolutely continuous or singular, using Hahn decomposition and Hellinger processes, with explicit formulas for their measures' components.

Contribution

It provides a novel characterization of absolute continuity and singularity of measures on filtered spaces via Hahn decomposition and Hellinger processes, including explicit formulas for measure components.

Findings

Hahn decomposition of measures on filtered spaces is characterized using Hellinger processes.

Explicit formulas for the norm of the absolutely continuous component are derived.

The relationship between measure restrictions at stopping times and their decompositions is established.

Abstract

Let $\mu$ and $\nu$ be fixed probability measures on a filtered space $(\Omega, {\cal F}, ({\cal F}_t)_{t\in {\bf R}^{+}})$. Denote by $\mu_T $ and $\nu_T $ (respectively, $\mu_{T-} $ and $\nu_{T-} $) the restrictions of the measures $\mu$ and $\nu$ on ${\cal F}_T $ (respectively, on ${\cal F}_{T-} $) for a stopping time $T$. We find the Hahn decomposition of $\mu_T $ and $\nu_T $ using the Hahn decomposition of the measures $\mu$, $\nu$, and the Hellinger process $h_t$ in the strict sense of order 1/2. The norm of the absolutely continuous component of $\mu_{T-} $ with respect to $\nu_{T-} $ is computed in terms of density processes and Hellinger integrals.