Comptes Rendus
no. 11-12
Probability Theory
A characterization of the set-indexed fractional Brownian motion by increasing paths
[Une caractérisation par chemins croissants du mouvement brownien fractionnaire indexé par des ensembles]

Présenté par : Marc Yor

Erick Herbin1 ; Ely Merzbach2
1 Dassault Aviation, 78, quai Marcel-Dassault, 92552 Saint-Cloud cedex, France
2 Department of Mathematics, Bar Ilan University, 52900 Ramat-Gan, Israel
Comptes Rendus. Mathématique, Volume 343 (2006) no. 11-12, pp. 767-772.

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.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2006.11.009

Erick Herbin 1 ; Ely Merzbach 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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Erick Herbin; Ely Merzbach. 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. https://comptes-rendus.academie-sciences.fr/mathematique/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] G. Ivanoff Set-indexed processes: distributions and weak convergence, Topics in Spatial Stochastic Processes, Lecture Notes in Mathematics, vol. 1802, Springer, 2003, pp. 85-126

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

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

  • Arthur Yosef Selected topics in the generalized mixed set-indexed fractional Brownian motion, Journal of Theoretical Probability, Volume 34 (2021) no. 3, pp. 1366-1381 | DOI:10.1007/s10959-021-01077-6 | Zbl:1496.60032
  • Arthur Yosef; Amos Baranes Karhunen-Loève expansion of a set indexed fractional Brownian motion, Statistics Probability Letters, Volume 156 (2020), p. 6 (Id/No 108629) | DOI:10.1016/j.spl.2019.108629 | Zbl:1453.60090
  • Arthur Yosef A group action on increasing sequences of set-indexed Brownian motions, Modern Stochastics. Theory and Applications, Volume 2 (2015) no. 2, pp. 185-198 | DOI:10.15559/15-vmsta31 | Zbl:1352.60116
  • Erick Herbin; Ely Merzbach Stationarity and self-similarity characterization of the set-indexed fractional Brownian motion, Journal of Theoretical Probability, Volume 22 (2009) no. 4, pp. 1010-1029 | DOI:10.1007/s10959-008-0180-8 | Zbl:1197.60040
  • Ely Merzbach; Arthur Yosef Set-indexed Brownian motion on increasing paths, Journal of Theoretical Probability, Volume 22 (2009) no. 4, pp. 883-890 | DOI:10.1007/s10959-008-0188-0 | Zbl:1181.60060
  • Arthur Yosef Set indexed strong martingales and path independent variation, Statistics Probability Letters, Volume 79 (2009) no. 8, pp. 1083-1088 | DOI:10.1016/j.spl.2008.12.014 | Zbl:1169.60007

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