Let X be a cadlag Lévy process with $X\_0=0$ and let $p$ be a real number in $[1,\infty)$. Then, the following are equivalent.


1): $X$ is $L^p$-integrable.


2): $X^\*\_t= \mathop{\sup}\_{0\leq s\leq t} |X\_s|$ is $L^p$ integrable.


I would like to see a reference such that I can refer to it this theorem.



Running supremmum of a Levy process
See Theorem 25.18 p168 in "Lévy Processes and Infinitely Divisible Distributions" by K-I Sato 


