Consider the stochastic integral:
$$X(t)=\int_0^tf(s-)dL(s)$$
with $L$ being a Lévy process (and $f$ properly chosen in order to let the SI exist).
Do there exists conditions on $f$, excluding the trivial one: $f$ is constant, such that $X$ is again a Lévy process?
In my particular interest, $L$ is a compound Poisson process. Do there exists conditions on $f$ such that $X$ is Lévy or even a compound Poisson process again?