Abstract:
In this work there are given necessary and sufficient conditions for the absolute continuity and equivalence (μξ≪μω, μω≪μξ, μξ∼μω) of a Wiener measure μω and a measure μξ corresponding to a process ξ of diffusion type with differential dξt=at(ξ)dt+dωt.
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 μξ and μω are investigated for the case when ξ 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 ξ for which μξ∼μω is a process of diffusion type.
Citation:
R. Sh. Liptser, A. N. Shiryaev, “On the absolute continuity of measures corresponding to processes of diffusion type relative to a Wiener measure”, Math. USSR-Izv., 6:4 (1972), 839–882
\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 Math. USSR-Izv.
\yr 1972
\vol 6
\issue 4
\pages 839--882
\mathnet{http://mi.mathnet.ru/eng/im2337}
\crossref{https://doi.org/10.1070/IM1972v006n04ABEH001903}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=312562}
\zmath{https://zbmath.org/?q=an:0267.60079}
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https://www.mathnet.ru/eng/im2337
https://doi.org/10.1070/IM1972v006n04ABEH001903
https://www.mathnet.ru/eng/im/v36/i4/p847
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