Lamperti transform of system of SDE's

Hello everyone,

I am considering a system of two coupled SDE's on the form:

$\displaystyle dX_t = f_{Xt}(X_t,Y_t,t)dt + g_{Xt}(X_t,t)dB_t^{(0)} \\ dY_t = f_{Yt}(X_t,Y_t,t)dt + g_{Yt}(Y_t,t)dB_t^{(1)}$

Where $\displaystyle dB_t^{(0)}$ and $\displaystyle dB_t^{(1)}$ are independent Brownian increments.
And want to Lamperti transform the system, i.e transform $\displaystyle dX_t$ and $\displaystyle dY_t$ such that the functions
$\displaystyle g_{Xt}(X_t,t)$ and $\displaystyle g_{Yt}(Y_t,t)$ becomes constants. All function are assumed sufficiently smooth.
I know how to do the transform if only considering one SDE.
Thus my question is can I transform each equation independently?

Because then the functions I need for transformation is just:

$\displaystyle L_{X_t}(x) = \int^x \frac{1}{g_{Xt}(v,t)} dv \\ L_{Y_t}(x) = \int^x \frac{1}{g_{Yt}(v,t)} dv$

Followed by applying Ito's lemma.

Thank you