Geodesic
On the absolute continuity of measures corresponding to processes of diffusion type relative to a~Wiener measure
Izvestiya. Mathematics , Tome 6 (1972) no. 4, pp. 839-882

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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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     author = {R. Sh. Liptser and A. N. Shiryaev},
     title = {On the absolute continuity of measures corresponding to processes of diffusion type relative to {a~Wiener} measure},
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     year = {1972},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1972_6_4_a8/}
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R. Sh. Liptser; A. N. Shiryaev. On the absolute continuity of measures corresponding to processes of diffusion type relative to a~Wiener measure. Izvestiya. Mathematics , Tome 6 (1972) no. 4, pp. 839-882. http://geodesic.mathdoc.fr/item/IM2_1972_6_4_a8/