Quelques approximations du temps local brownien

Blandine Bérard Bergery; Pierre Vallois

doi:10.1016/j.crma.2007.05.007

Probabilités

Quelques approximations du temps local brownien
[Some Brownian local time approximations]

Presented by: Marc Yor

Blandine Bérard Bergery ¹; Pierre Vallois ¹

¹ Université Henri-Poincaré, institut de mathématiques Elie-Cartan, B.P. 239, 54506 Vandœuvre-lès-Nancy cedex, France

Comptes Rendus. Mathématique, Volume 345 (2007) no. 1, pp. 45-48.

Abstract
Résumé

We give some approximations of the local time process ${(L_{t}^{x})}_{t ⩾ 0}$ at level x of the real Brownian motion $(X_{t})$ . We prove that $\frac{2}{ϵ} \int_{0}^{t} X_{(u + ϵ) \land t}^{+} 1_{{X_{u} ⩽ 0}} d u + \frac{2}{ϵ} \int_{0}^{t} X_{(u + ϵ) \land t}^{-} 1_{{X_{u} > 0}} d u$ and $\frac{4}{ϵ} \int_{0}^{t} X_{u}^{-} 1_{{X_{(u + ϵ) \land t} > 0}} d u$ converge in the ucp sense to $L_{t}^{0}$ , as $ϵ \to 0$ . We show that $\frac{1}{ϵ} \int_{0}^{t} (1_{{x < X_{s + ϵ}}} - 1_{{x < X_{s}}}) (X_{s + ϵ} - X_{s}) d s$ goes to $L_{t}^{x}$ in $L^{2} (Ω)$ as $ϵ \to 0$ , and that the rate of convergence is of order $ϵ^{α}$ , for any $α < \frac{1}{4}$ .

On définit plusieurs approximations du processus des temps locaux ${(L_{t}^{x})}_{t ⩾ 0}$ au niveau x du mouvement brownien réel $(X_{t})$ . En particulier, on montre que $\frac{2}{ϵ} \int_{0}^{t} X_{(u + ϵ) \land t}^{+} 1_{{X_{u} ⩽ 0}} d u + \frac{2}{ϵ} \int_{0}^{t} X_{(u + ϵ) \land t}^{-} 1_{{X_{u} > 0}} d u$ et $\frac{4}{ϵ} \int_{0}^{t} X_{u}^{-} 1_{{X_{(u + ϵ) \land t} > 0}} d u$ convergent au sens ucp vers $L_{t}^{0}$ , lorsque $ϵ \to 0$ . D'autre part, on montre que $\frac{1}{ϵ} \int_{0}^{t} (1_{{x < X_{s + ϵ}}} - 1_{{x < X_{s}}}) (X_{s + ϵ} - X_{s}) d s$ converge vers $L_{t}^{x}$ dans $L^{2} (Ω)$ et que la vitesse de convergence est d'ordre $ϵ^{α}$ , pour tout $α < \frac{1}{4}$ .

Received: 2006-08-30
Accepted: 2007-05-04
Published online: 2007-06-21

DOI: 10.1016/j.crma.2007.05.007

Author's affiliations:

Blandine Bérard Bergery ¹; Pierre Vallois ¹

¹ Université Henri-Poincaré, institut de mathématiques Elie-Cartan, B.P. 239, 54506 Vandœuvre-lès-Nancy cedex, France

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     author = {Blandine B\'erard Bergery and Pierre Vallois},
     title = {Quelques approximations du temps local brownien},
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     pages = {45--48},
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     number = {1},
     year = {2007},
     doi = {10.1016/j.crma.2007.05.007},
     language = {fr},
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Blandine Bérard Bergery; Pierre Vallois. Quelques approximations du temps local brownien. Comptes Rendus. Mathématique, Volume 345 (2007) no. 1, pp. 45-48. doi : 10.1016/j.crma.2007.05.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.05.007/

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