# Characteristic Exponent of Levy Process

I'm reading a chapter of a book on Levy processes and it states:

"Any Levy process $X$ enjoys the following property: For all $t\geq0$ $$\mathbb{E}[e^{iu X_t}]=e^{t\phi(u)},$$ where $\phi$ is the characterstic function of $X$ of the process $X$".

I read somewhere else that this is a consequence of stationary increment property of Levy processes. Can somoneone provide me with more insight on this? I want to understand why this holds

If you denote the left side by $f(t)$ then $f(t+s)=f(t)f(s)$. This is an easy consequence of the fact that the process has stationary independent increments. Since $f$ is a continuous function and $f(0)=1$ we can show that $f$ must of the type specified.