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Izv. Akad. Nauk SSSR Ser. Mat., 1972, Volume 36, Issue 4, Pages 847–889 (Mi izv2337)  

This article is cited in 10 scientific papers (total in 11 papers)

On the absolute continuity of measures corresponding to processes of diffusion type relative to a Wiener measure

R. Sh. Liptser, A. N. Shiryaev


Abstract: In this work there are given necessary and sufficient conditions for the absolute continuity and equivalence ($\mu_\xi\ll\mu_\omega$, $\mu_\omega\ll\mu_\xi$, $\mu_\xi\sim\mu_\omega$) of a Wiener measure $\mu_\omega$ and a measure $\mu_\xi$ corresponding to a process $\xi$ of diffusion type with differential $d\xi_t=a_t(\xi) dt+d\omega_t$.
The densities (the Radon–Nikodým derivatives) of one measure with respect to the other are found. Questions of the absolute continuity and equivalence of measures $\mu_\xi$ and $\mu_\omega$ are investigated for the case when $\xi$ is an Ito process. Conditions under which an Ito process is of diffusion type are derived. It is proved that (up to equivalence) every process $\xi$ for which $\mu_\xi\sim\mu_\omega$ is a process of diffusion type.

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English version:
Mathematics of the USSR-Izvestiya, 1972, 6:4, 839–882

Bibliographic databases:

UDC: 519.2
MSC: Primary 60J60, 60G30; Secondary 28A40
Received: 17.09.1971

Citation: R. Sh. Liptser, A. N. Shiryaev, “On the absolute continuity of measures corresponding to processes of diffusion type relative to a Wiener measure”, Izv. Akad. Nauk SSSR Ser. Mat., 36:4 (1972), 847–889; Math. USSR-Izv., 6:4 (1972), 839–882

Citation in format AMSBIB
\Bibitem{LipShi72}
\by R.~Sh.~Liptser, A.~N.~Shiryaev
\paper On the absolute continuity of measures corresponding to processes of diffusion type relative to a~Wiener measure
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1972
\vol 36
\issue 4
\pages 847--889
\mathnet{http://mi.mathnet.ru/izv2337}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=312562}
\zmath{https://zbmath.org/?q=an:0267.60079}
\transl
\jour Math. USSR-Izv.
\yr 1972
\vol 6
\issue 4
\pages 839--882
\crossref{https://doi.org/10.1070/IM1972v006n04ABEH001903}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. D.A Dawson, “Stochastic evolution equations and related measure processes”, Journal of Multivariate Analysis, 5:1 (1975), 1  crossref
    2. G. A. Melnichenko, “Struktura protsessov, absolyutno nepreryvnykh otnositelno gaussovskogo”, UMN, 32:1(193) (1977), 197–198  mathnet  mathscinet  zmath
    3. H. Ito, “Probabilistic Construction of Lagrangean of Diffusion Process and Its Application”, Progress of Theoretical Physics, 59:3 (1978), 725  crossref
    4. Yu. M. Kabanov, R. Sh. Liptser, A. N. Shiryaev, “Absolute continuity and singularity of locally absolutely continuous probability distributions. II”, Math. USSR-Sb., 36:1 (1980), 31–58  mathnet  crossref  mathscinet  zmath  isi
    5. Ishwar V. Basawa, B.L.S. Prakasa Rao, “Asymptotic inference for stochastic processes”, Stochastic Processes and their Applications, 10:3 (1980), 221  crossref
    6. Philippe Blanchard, Piotr Garbaczewski, “Natural boundaries for the Smoluchowski equation and affiliated diffusion processes”, Phys Rev E, 49:5 (1994), 3815  crossref  mathscinet  isi
    7. M. Jerschow, “Infinite-dimensional Wiener processes with drift”, Stochastic Processes and their Applications, 52:2 (1994), 229  crossref
    8. F. Klebaner, R. Liptser, “When a stochastic exponential is a true martingale. Extension of the Beneš method”, Theory Probab. Appl., 58:1 (2014), 38–62  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    9. Johannes Ruf, “A new proof for the conditions of Novikov and Kazamaki”, Stochastic Processes and their Applications, 123:2 (2013), 404  crossref
    10. Carole Bernard, Zhenyu Cui, Don McLeish, “ON THE MARTINGALE PROPERTY IN STOCHASTIC VOLATILITY MODELS BASED ON TIME-HOMOGENEOUS DIFFUSIONS”, Mathematical Finance, 2014, n/a  crossref
    11. V. M. Abramov, B. M. Miller, E. Ya. Rubinovich, P. Yu. Chiganskii, “Razvitie teorii stokhasticheskogo upravleniya i filtratsii v rabotakh R. Sh. Liptsera”, Avtomat. i telemekh., 2020, no. 3, 3–13  mathnet  crossref
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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