Separating times for measures on filtered spaces

M. A. Urusov ; A. S. Cherny

Teoriâ veroâtnostej i ee primeneniâ, Tome 48 (2003) no. 2, pp. 416-427

Résumé

We introduce the notion of a separating time for a pair of measures $P$ and $\widetilde{P}$ on a filtered space. This notion is convenient for describing the mutual arrangement of $P$ and $\widetilde{P}$ from the viewpoint of the absolute continuity and singularity. Furthermore, we find the explicit form of the separating time for the case, where $P$ and $\widetilde{P}$ are distributions of Lévy 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 $P$ and $\widetilde{P}$.

Détail

Keywords: separating time, local absolute continuity, absolute continuity, singularity, Lévy processes, stochastic differential equations, Bessel processes.