Probability Theory
A characterization of the set-indexed fractional Brownian motion by increasing paths
Comptes Rendus. Mathématique, Volume 343 (2006) no. 11-12, pp. 767-772.

We prove that a set-indexed process is a set-indexed fractional Brownian motion if and only if its projections on all the increasing paths are one-parameter time changed fractional Brownian motions. As an application, we present an integral representation for such processes.

On montre qu'un processus stochastique est un mouvement brownien fractionnaire indexé par des ensembles si et seulement si ses projections sur tous les chemins croissants sont des mouvements browniens fractionnaires à paramètres réels changés de temps. On applique ce résultat à la définition d'une représentation intégrale pour de tels processus.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2006.11.009
Herbin, Erick 1; Merzbach, Ely 2

1 Dassault Aviation, 78, quai Marcel-Dassault, 92552 Saint-Cloud cedex, France
2 Department of Mathematics, Bar Ilan University, 52900 Ramat-Gan, Israel
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Herbin, Erick; Merzbach, Ely. A characterization of the set-indexed fractional Brownian motion by increasing paths. Comptes Rendus. Mathématique, Volume 343 (2006) no. 11-12, pp. 767-772. doi : 10.1016/j.crma.2006.11.009. http://www.numdam.org/articles/10.1016/j.crma.2006.11.009/

[1] E. Herbin, E. Merzbach, A set-indexed fractional Brownian motion, J. Theoret. Probab. (2006), in press

[2] E. Herbin, E. Merzbach, The multiparameter fractional Brownian motion, in: Proceedings of VK60 Math Everywhere Workshop, 2006, in press

[3] Ivanoff, G. Set-indexed processes: distributions and weak convergence, Topics in Spatial Stochastic Processes, Lecture Notes in Mathematics, vol. 1802, Springer, 2003, pp. 85-126

[4] Ivanoff, G.; Merzbach, E. Set-Indexed Martingales, Chapman & Hall/CRC, 2000

[5] Rogers, C.A. Hausdorff Measures, Cambridge Univ. Press, 1970

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