I don't think Girsanov's formula works when the volatilities are different between the P measure and P* measure. P and P* will be singular with respect to each other.
Please see Prof. Goodman's class notes on page 11 at http://www.math.nyu.edu/faculty/goodman/teaching/StochCalc2012/notes/Week10.pdf .
Also, from [ http://ocw.mit.edu/courses/sloan-school-of-management/15-450-analytics-of-finance-fall-2010/lecture-notes/MIT15_450F10_lec02.pdf ] page 54:
a probability measure assigns relative likelihood to different trajectories of the Brownian motion. Variance of the Ito process can be recovered from the shape of a single trajectory (quadratic variation), so it does not depend on the relative likelihood of the trajectories, hence, does not depend on the choice of the probability measure.