The Annals of Probability

Controle de la Norme $H^p$ D'Une Martingale par des Maximums de Temps Locaux

Christophe Leuridan

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Abstract

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.

Article information

Source
Ann. Probab., Volume 23, Number 3 (1995), 1289-1299.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176988184

Digital Object Identifier
doi:10.1214/aop/1176988184

Mathematical Reviews number (MathSciNet)
MR1349172

Zentralblatt MATH identifier
0833.60049

JSTOR
links.jstor.org

Subjects
Primary: 60G44: Martingales with continuous parameter
Secondary: 60J55: Local time and additive functionals 60J65: Brownian motion [See also 58J65]

Keywords
Martingales Brownian motion local times maximums of local times $H^p$ norms

Citation

Leuridan, Christophe. Controle de la Norme $H^p$ D'Une Martingale par des Maximums de Temps Locaux. Ann. Probab. 23 (1995), no. 3, 1289--1299. doi:10.1214/aop/1176988184. https://projecteuclid.org/euclid.aop/1176988184


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