risk neutral measure - Version of Girsanov theorem with changing volatility

Is there a version of Girsanov theorem when the volatility is changing?

For example Girsanov theorem states that Radon Nikodym (RN) derivative for a stochastic equation is used to transform the expectation where the sampling is done in one mesaure to an expectation where sampling is done in another measure.

Let's see an example

$dX_t(w) = f(X_t(w))dt + \sigma(X_t(w))dW_t^P(w)$ in P measure.

In P* measure, drift is $f^{*}(X_t(w))$. We multiply the internals of expectation in P measure with RN derivative to get expectation of X in P* measure

$E^{P^*}[X] = E^P[X \frac{dP^*}{dP}]$

where

$\frac{dP^*}{dP}=e^{-0.5 \int (\frac{ f^{*}(X_s(w)) - f(X_s(w))}{\sigma(X_s(w))})^2ds + \int \frac{ f*(X_s(w)) - f(X_s(w))}{\sigma(X_s(w))} dW_s^P(w)}$

What I am looking for is in P* measure, not only drift but also the volatility changes

$dX_t(w) = f^{*}(X_t(w))dt + \sigma^{*}(X_t(w))dW_t^P(w)$

Then what is $\frac{dP^*}{dP}$?