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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.

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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).

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  • $\begingroup$ That is definitely interesting. I obtained some similar relation for the case $k=4$. The example of $\alpha$-stable processes suggested that this could be generalized for non-even (possible fractional) $k$. If you have any suggestions of source for such a question, I would appreciate. Whatever, thanks a lot for the paper. $\endgroup$ – Goulifet Apr 22 '14 at 12:26

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