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Mat. Sb. (N.S.), 1978, Volume 107(149), Number 3(11), Pages 364–415 (Mi msb2679)  

This article is cited in 47 scientific papers (total in 47 papers)

Absolute continuity and singularity of locally absolutely continuous probability distributions. I

Yu. M. Kabanov, R. Sh. Liptser, A. N. Shiryaev


Abstract: 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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English version:
Mathematics of the USSR-Sbornik, 1979, 35:5, 631–680

Bibliographic databases:

UDC: 519.2
MSC: Primary 60G30, 60G45, 60H05; Secondary 28A40, 60G25, 60G40
Received: 11.01.1978

Citation: Yu. M. Kabanov, R. Sh. Liptser, A. N. Shiryaev, “Absolute continuity and singularity of locally absolutely continuous probability distributions. I”, Mat. Sb. (N.S.), 107(149):3(11) (1978), 364–415; Math. USSR-Sb., 35:5 (1979), 631–680

Citation in format AMSBIB
\Bibitem{KabLipShi78}
\by Yu.~M.~Kabanov, R.~Sh.~Liptser, A.~N.~Shiryaev
\paper Absolute continuity and singularity of locally absolutely continuous probability distributions.~I
\jour Mat. Sb. (N.S.)
\yr 1978
\vol 107(149)
\issue 3(11)
\pages 364--415
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\mathscinet{http://www.ams.org/mathscinet-getitem?mr=515738}
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\transl
\jour Math. USSR-Sb.
\yr 1979
\vol 35
\issue 5
\pages 631--680
\crossref{https://doi.org/10.1070/SM1979v035n05ABEH001615}
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