There is an implication that I am not able to find by myself in my lecture notes.
I consider $X_t$ a cadlag process with values in $\mathbb{R}_{+}$ such that $X_t$ is locally integrable (with respect to $t$) and integrable with respect to $\omega$. We define the function $\phi(s) = \mathbb{E}[X_s]$ and I have the following functional equation for $0<a<b$ and $\lambda$ fixed
$$ \phi(b) = \phi(a) + \lambda(e^{iu}-1)\mathbb{E}[\int_{a}^{b} X_s ds] $$
This implies the following equality
$$ \phi(t) = \phi(a)e^{\lambda(e^{iu}-1)(t-a)} $$
for all $t\geq a$. I do not see how to find this implication, I guess there is (maybe) a link with the solution of an ODE when looking at the exponential term ? I would like to have some hint please rather than a full answer please.