This is a question I got while reading the proof of Lemma 4 from the following post: https://almostsuremath.com/2010/06/16/continuous-processes-with-independent-increments/
Here, let $X$ be a continuous $d$-dimensional process with independent increments and $\psi:\mathbb{R}_+ \times \mathbb{R}^d \to \mathbb{C}$ be the unique continuous function with $\psi(0,a)=0$ and $$E[e^{ia\cdot (X_t - X_0)}]=e^{\psi(t,a)}.$$ Then we know that $e^{ia\cdot X_t - \psi(t,a)}$ is a martingale for each fixed $a\in \mathbb{R}^d$.
Now, set $Y_t = ia\cdot (X_t - X_0) - \psi_t(a)$. Then, $U:= \exp(Y)$ is the martingale $e^{ia\cdot X_t - \psi(t,a)}$, and hence a semimartingale.
The author then says that by Ito's lemma, $Y=\log(U)$ is a semimartingale.
From the Ito's lemma statement the author gives, this requires $U$ to take values in some open subset of $\mathbb{C}$ on which $\log$ is twice differentiable. For complex logarithms, this would be the case if $U$ takes values in $\mathbb{C}$ minus some branch cut. But how do we ensure this?
The author just states that there is no problem in applying Ito's lemma here because although the logarithm is not a well-defined twice differentiable function everywhere on $\mathbb{C}^\times$, this is true locally (actually, on any half plane).
I would greatly appreciate if anyone could explain how the domain of the complex logarithm does not become an issue here.
