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.
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@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 = {https://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 - https://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 https://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, Tome 343 (2006) no. 11-12, pp. 767-772. doi : 10.1016/j.crma.2006.11.009. https://www.numdam.org/articles/10.1016/j.crma.2006.11.009/
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