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.
Accepted:
Published online:
@article{CRMATH_2006__343_11-12_767_0, author = {Herbin, Erick and Merzbach, Ely}, title = {A characterization of the set-indexed fractional {Brownian} motion by increasing paths}, journal = {Comptes Rendus. Math\'ematique}, pages = {767--772}, publisher = {Elsevier}, volume = {343}, number = {11-12}, year = {2006}, doi = {10.1016/j.crma.2006.11.009}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.crma.2006.11.009/} }
TY - JOUR AU - Herbin, Erick AU - Merzbach, Ely TI - A characterization of the set-indexed fractional Brownian motion by increasing paths JO - Comptes Rendus. Mathématique PY - 2006 SP - 767 EP - 772 VL - 343 IS - 11-12 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2006.11.009/ DO - 10.1016/j.crma.2006.11.009 LA - en ID - CRMATH_2006__343_11-12_767_0 ER -
%0 Journal Article %A Herbin, Erick %A Merzbach, Ely %T A characterization of the set-indexed fractional Brownian motion by increasing paths %J Comptes Rendus. Mathématique %D 2006 %P 767-772 %V 343 %N 11-12 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2006.11.009/ %R 10.1016/j.crma.2006.11.009 %G en %F CRMATH_2006__343_11-12_767_0
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] Set-indexed processes: distributions and weak convergence, Topics in Spatial Stochastic Processes, Lecture Notes in Mathematics, vol. 1802, Springer, 2003, pp. 85-126
[4] Set-Indexed Martingales, Chapman & Hall/CRC, 2000
[5] Hausdorff Measures, Cambridge Univ. Press, 1970
Cited by Sources: