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

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

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
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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
2. G. A. Melnichenko, “Struktura protsessov, absolyutno nepreryvnykh otnositelno gaussovskogo”, UMN, 32:1(193) (1977), 197–198
3. H. Ito, “Probabilistic Construction of Lagrangean of Diffusion Process and Its Application”, Progress of Theoretical Physics, 59:3 (1978), 725
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
5. Ishwar V. Basawa, B.L.S. Prakasa Rao, “Asymptotic inference for stochastic processes”, Stochastic Processes and their Applications, 10:3 (1980), 221
6. Philippe Blanchard, Piotr Garbaczewski, “Natural boundaries for the Smoluchowski equation and affiliated diffusion processes”, Phys Rev E, 49:5 (1994), 3815
7. M. Jerschow, “Infinite-dimensional Wiener processes with drift”, Stochastic Processes and their Applications, 52:2 (1994), 229
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
9. Johannes Ruf, “A new proof for the conditions of Novikov and Kazamaki”, Stochastic Processes and their Applications, 123:2 (2013), 404
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
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
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