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Let $(X_t)$ be an Ito diffusion with speed $(V_t)$, under a probability measure P. Could there exist a change of measure to a probability measure Q, with Q ~ P, under which $(X_t)$ is an Ito diffusion with a different speed $(V'_t)$?

I've figured out a way such that QV(X) = $\displaystyle\int_{0}^{T} V_t^2 dt.$ With probability 1. Also by Radon-Nikodym derivative of Q with respect to P such that $T = \frac{dQ}{dP}$. So we can get $\frac{dQ}{dP}$ = $\frac{\displaystyle\int_{0}^{T} (V'_t)^2 dt}{\displaystyle\int_{0}^{T} V_t^2 dt}$ But I'm stuck after that. If anyone can give any ideas of how to continue, or if the way I did was incorrect. Thanks

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