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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 [ ...

The Girsanov theorem applies to any compatible change of measure, including a volatility change. The version you have written above is a simplified version for drift changes only, but if you look in any good stochastic calculus book, you will see that full version just requires that you be able to compute the cross-variation of the two processes.