# Ito Diffusion with Change of Measure

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