I feel there is some ambiguity for the definition of the exponential martingale for a levy process which I do not understand.
For a Lévy process it can be shown that $E[e^{iuX_t}]=e^{t\eta(u)}$, where $\eta(u)$ is the levy symbol. It can then be shown that $Z_t(u)=e^{iuX_t-t\eta(u)}$ is a martingale. It is called the exponential-martingale.
Another way I see in another book is that we can define $\phi_t(u)=E[e^{iuX_t}]$, and show that this is never 0. Then we can define the expnential martingale as $Z'_t(u)=\frac{e^{iuX_t}}{\phi_t(u)}$.
Now these are the same if t is constant, because then we have: $Z'_t(u)=\frac{e^{iuX_t}}{\phi_t(u)}=\frac{e^{iuX_t}}{E[e^{iuX_t}]}=e^{iux_t-t\eta(u)}=Z_t(u)$. So both definitions give the same martingale.
But looking at the r.v. $Z_\tau(u)$ and $Z'_\tau(u)$. The first definition gives $Z_\tau(u)=e^{iuX_t-\tau\eta(u)}$, the second definiton gives $Z'_\tau=e^{iuX_\tau}/E[e^{iuX_\tau}]$. Now they may not be the same because $E[e^{iuX_\tau}]$ is constant, but $e^{-\tau\eta(u)}$ is a r.v.
It can be shown that for any martingale then the stopped process is also a martingale. Does this mean that both $Z_{t\wedge\tau}(u)$, and $Z'_{t\wedge \tau}(u)$ are martingales? Are they unequal, but both are martingales?
Rather, $Z'_\tau(u) = e^{iuX_\tau}/\phi_\tau(u)$. Do notice that the random variable $\omega\mapsto\phi_{\tau(\omega)}(u)$ appearing in the denominator of this fraction is the composition of the function $t\mapsto \phi_t(u)$ with the random variable $\omega\mapsto \tau(\omega)$. This is not the same as the complex number $E[e^{iuX_\tau}]$.
$\{Z_t(u)\}_{t\ge 0}$ and $\{Z'_t(u)\}_{t\ge 0}$ are identical processes, as are their stopped versions.