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I want to know if this system of SDE: $$dX_{t}=b(X_{t})dt+\sigma( X_{t}) dB_{t}$$ $$dY_{t}=b_{0}(Y_{t})dt+\sigma( Y_{t}) dB_{t}$$ at what conditions on drifts the law of diffusions are absolutly continu each other? What's happen when coefficients of diffusions are different and same drifts? NB: B is standard brownian motion.

Thanks.

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  • $\begingroup$ Drifts don't matter, I think. Laws of diffusions with constant diffusion coefficients $\sigma_1$ and $\sigma_2$ are absolutely continuous with each other iff $\sigma_1=\sigma_2$ (just as Brownian processes). $\endgroup$
    – Jean Duchon
    Commented Jun 11, 2016 at 10:44
  • $\begingroup$ @Jean thanks, i have read somewhere that if diffusion coefficients are the same function there is absolute continuity. and if there are different we loss absolute continuity. It's like you said. Is it condition for existence of solutions? $\endgroup$
    – user88853
    Commented Jun 11, 2016 at 23:42

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