Moment estimates for Lévy Processes

Harald Luschgy (1) , Gilles Pagès (2)

1 FB IV-Mathematik
2 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

For real Lévy processes $(X_t)_{t \geq 0}$ having no Brownian component with Blumenthal-Getoor index $\beta$, the estimate $\E \sup_{s \leq t} | X_s - a_p s |^p \leq C_p t$ for every $t \!\in [0,1]$ and suitable $a_p \!\in \R$ has been established by Millar \cite{MILL} for $\beta < p \leq 2$ provided $X_1 \!\in L^p$. We derive extensions of these estimates to the cases $p > 2$ and $p \leq\beta$.

Keywords: Lévy process increment; Lévy measure; \alpha-stable process; Normal Inverse Gaussian process; tempered stable process; Meixner process.; Meixner process

Domains: Probability [math.PR]

Harald Luschgy, Gilles Pagès. Moment estimates for Lévy Processes. Electronic Journal of Probability, 2008, 13, 422-434 : http://dx.doi.org/10.1214/ECP.v13-1397. ⟨10.1214/ECP.v13-1397⟩. ⟨hal-00085213⟩