Let $B$ be a brownian motion starting at 0. We denote by $L^\ast_t = \max_{x\in\mathbb{R}} L^\ast_t$ the maximum of local times at time $t$. The Barlow-Yor inequalities tell us that for every $p > 0$, there are constants $C_p > c_p > 0$ such that for every stopping time $\tau$, $c_p\mathbb{E}\lbrack\tau^{p/2}\rbrack \leq \mathbb{E}\lbrack L^{\ast p}_\tau\rbrack \leq C_p\mathbb{E}\lbrack\tau^{p/2}\rbrack.$ Given a fixed closed set $F \subset \mathbb{R}$, we give a condition on $F$ which is necessary and sufficient to derive similar inequalities with $\max_{x\in F}L^x_\tau$ instead of $L^\ast_\tau$ and we prove various related results.
Publié le : 1995-07-14
Classification:
Martingales,
Brownian motion,
local times,
maximums of local times,
$H^p$ norms,
60G44,
60J55,
60J65
@article{1176988184,
author = {Leuridan, Christophe},
title = {Controle de la Norme $H^p$ D'Une Martingale par des Maximums de Temps Locaux},
journal = {Ann. Probab.},
volume = {23},
number = {3},
year = {1995},
pages = { 1289-1299},
language = {fr},
url = {http://dml.mathdoc.fr/item/1176988184}
}
Leuridan, Christophe. Controle de la Norme $H^p$ D'Une Martingale par des Maximums de Temps Locaux. Ann. Probab., Tome 23 (1995) no. 3, pp. 1289-1299. http://gdmltest.u-ga.fr/item/1176988184/