Separating Times for Measures on Filtered Spaces
Alexander S. Cherny, Mikhail Aleksandrovich Urusov · Theory of Probability and Its Applications · 2004
We introduce the notion of a {\it separating time} for a pair of measures~${\bf P}$ and~${\widetilde{\bf P}}$ on a filtered space. This notion is convenient for describing the mutual arrangement of~${\bf P}$ and~${\widetilde{\bf P}}$ from the viewpoint of the absolute continuity and singularity. Furthermore, we find the explicit form of the separating time for the case, where ${\bf P}$ and ${\widetilde{\bf P}}$ are distributions of Levy processes, solutions of stochastic differential equations, and distributions of Bessel processes. The obtained results yield, in particular, the criteria for the local absolute continuity, absolute continuity, and singularity of~${\bf P}$ and~${\widetilde{\bf P}}$.