I'm new to the world of Lévy processes. I understand that all Lévy processes $X = (X_t)_{t\geq 0}$ have the following property: For $t\geq 0$, there exists a characteristic function $\phi$ defined by $\mathbb{E}\Big[e^{i\theta X_t}\Big] = e^{-t\phi(\theta)},\quad\theta \in \mathbb{R}$.
I'm currently reading a paper where the following Lévy process is used:
$$X_t = \bigg(r - \delta - \frac{\sigma^2}{2} - \lambda \mathbb{E}[Z_1]\Bigg)t + \sigma W_t + \sum_{k=1}^{N_t}In(Z_k+ 1)$$
where $r$, $\delta$, $\sigma$ are constants. $N_t$ is a Poisson process, and $Z_k's$ are i.i.d randon variables. The paper then uses the following definition for the characteristic function (and Laplace exponent, saying they are equivalent):
$$\mathbb{E}\Big[e^{\theta X_t}\Big] = e^{t\phi(\theta)},\quad\theta \in \mathbb{R}.$$
What properties does $X$ possess so that this definition of characteristic function can be used?? I feel like I am missing something very obvious. Many thanks!
The process you mentioned is a special Lévy process, the pure discontinuous part of this Lévy process is a compound Poisson process. May be the Chapter 2 of following book could provide related information about it: A. E. Kyprianou, Fluctuations of Lévy Processes with Applications, 2nd Ed., Springer,(2014).