I was able to show the following:
$X$ a semimartingale, $X_0=0$ then the SDE
$ dZ_t=Z_tdX_t$
with $ Z_0=1$ has the unique solution $Z_t:=\exp{(X_t-\frac{1}{2}\langle X\rangle_t)}$.
I was able to show that this is indeed the a solution. However I got stuck at uniqueness. What I did so far: Suppose there are two solution, $Z,Z^'$. Apply Itô to $f(x,y):=\frac{x}{y}$ and with the $\mathbb{R}^2$ semimartingale $(Z,Z^')$. After some calculations, I'm at the point:
$df(Z_t,Z^'_t)=\frac{Z^'_t}{Z^3_t}d\langle Z\rangle_t -\frac{1}{Z^2_t}d\langle Z,Z^'\rangle_t$
The last two term should be equal, then uniqueness follows. Now why is this, i.e. why is it true that
$d\langle Z\rangle_t = (Z_t)^2d\langle X\rangle_t$
and
$ d\langle Z,Z^' \rangle_t = Z^'Zd\langle X\rangle_t$
Thank you for your help.