Concerning Jump process (Lévy process)

Question

Consider $X= \left( X_t \right)_{t\geq 0}$ is a Lévy process whose characteristic triplet is $\left( \gamma, \sigma ^2, \nu \right)$ and where its Lévy measure is

$$ \nu \left( dx\right) = A \sum_{n=1} ^{\infty} p^n \delta_{-n} \left( dx \right) + Bx^{\beta-1}\left( 1+x \right)^{-\alpha -\beta}e^{-\lambda x } \mathbf{1}_{\left ]0,+\infty \right[}\left( x\right)dx.$$

I'd like to know how to show that $Z_t = Z_0 \exp\left( \mu t + X_t \right)$ is well defined and admits first and second order moments.

I'm kind of lost here. I don't see what is the problem with this definition. Could someone please enlighten me ?

Must I show that $Z_t < \infty \ a.s.$ ?

Or maybe aplly Itô-Lévy lemma for derive the SDE $Z_t$ satisfies and so conclude that it's well defined as the unique strong solution of this SDE?

Or maybe another thing I've not even think about?

Answer 1

I don't think there should be any problem with the definition of $Z_t$, but of course showing that moments are finite can be tricky. You are trying to take exponential moments of $X_t$ and so you need fast enough exponential decay for the (right) tails of $X_t$. I'd need to review some stuff on Levy triples to be sure, but I suspect that the $e^{-\lambda x}$ term in the Levy measure $\nu$ implies that $Z_t$ has finite first moment if $\lambda>1$ and finite second moment if $\lambda > 2$.