Geodesic
Absolute continuity and singularity of locally absolutely continuous probability distributions. I
Sbornik. Mathematics, Tome 35 (1979) no. 5, pp. 631-680 
Yu. M. Kabanov; R. Sh. Liptser; A. N. Shiryaev. Absolute continuity and singularity of locally absolutely continuous probability distributions. I. Sbornik. Mathematics, Tome 35 (1979) no. 5, pp. 631-680. http://geodesic.mathdoc.fr/item/SM_1979_35_5_a2/
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     title = {Absolute continuity and singularity of locally absolutely continuous probability {distributions.~I}},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1979_35_5_a2/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

Let $(\Omega,\mathscr F)$ be a measurable space provided with a nondecreasing family of $\sigma$-algebras ($\mathscr F_t)_{t\geqslant0}$ with $\mathscr F=\bigvee_{t\geqslant0}\mathscr F_t$ and $\widetilde{\mathsf P}$ and $\mathsf P$ two locally absolutely continuous probability measures on $(\Omega,\mathscr F)$, i.e., such that $\widetilde{\mathsf P}_t\ll\mathsf P_t$ for $t\geqslant0$ ($\widetilde{\mathsf P}_t$ and $\mathsf P_t$ are the restrictions of $\widetilde{\mathsf P}$ and $\mathsf P$ to $\mathscr F_t$). One asks when $\widetilde{\mathsf P}\ll \mathsf P$ or $\widetilde{\mathsf P}\perp\mathsf P$. An answer to this question is given in terms of the convergence set of a certain increasing predictable process constructed for the martingale $\mathfrak Z=(\mathfrak Z_t,\mathscr F_t,\mathsf P)$ with $\mathfrak Z_t=d\widetilde{\mathsf P}_t/d\mathsf P_t$. Actually, the somewhat more general situation of $\theta$-local absolute continuity of measures is studied. The proof of the fundamental theorem is based on a series of results that are of independent interest. In § 2 the theory of integration with respect to random measures is developed. § 4 deals with the convergence sets of semimartingales, and § 5 with the transformation of the predictable characteristics of a semimartingale under a locally absolutely continuous change of measure. Sufficient conditions are given in § 7 for the uniform integrability of nonnegative local martingales. Bibliography: 24 titles.

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