On the moments of Lévy processes

Question

For a Brownian motion $B_t$, the evolution of the moments with $t$ obeys the simple rule: $$\mathbb{E}[|B_t|^p] = \kappa_p |t|^{p/2},$$ with $\kappa_p<\infty$. The proof only requires to remark that the random variables $\frac{B_t}{\sqrt{t}}$ are Gaussian with variance 1.

I am interested to know if, more generally, the question was studied for Lévy processes. Of course, in the general case, the moments can be infinite. When they are not, do we have an estimation of the evolution of $\mathbb{E}[|X_t|^p]$ for a Lévy process $X_t$, or at least some bounds?

Thank you for your attention.

Answer 1

In this paper (Proposition 2.3.), it is proved that if $k$ is an even, positive integer and $(L(t))_t$ is a Levy process with finite $k$-th moment, i.e. $\mathbb{E}\left\|L(1)\right\|^k<\infty$, then there exist real numbers $m_1,\ldots,m_k$ such that $$ \mathbb{E}\left\|L(t)\right\|^k=m_1t+m_2t^2+\ldots+m_kt^k,\quad t\geq 0. $$

It is not assumed that the Levy process has mean zero, hence the maximal exponent $k$ (rather than $p/2$ for your mean-zero Brownian motion).