Exponential Levy process

Question

We assume that the stochastic process L is a Levy process with the predictable characteristics triplet $(b,c,\nu)$. Which integrability conditions we should assume for the new stochastic process $$S=e^L$$ to be a semimartingale??

Answer 1

Since every Lévy process $(L_t)_{t\ge 0}$ is a semimartingale, you can take Itô's formula (see e.g. Protter Chpt. II) with the function $x\mapsto \text{e}^x$ (as @bmo said) to conclude that the stochastic process $$ (\text{e}^{L_t})_{t\ge 0} $$ is a semimartingale and no further moment assumptions on the Lévy process are necessary!