2007/06/23 by Érick Herbin, Erick Herbin, Herbin, Erick +2
Economics, Econometrics and Finance · Mathematics · #60G10 #60G15 #60G17 #60G18 #Complex Systems and Time Series Analysis #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and financial applications #math.PR #msc:60G10 #msc:60G15 #msc:60G17 #msc:60G18
paper · pdf · doi:10.48550/arxiv.0706.3472
18 pages
openalex publication_date 2007/06/23 · arxiv created 2008/07/09 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The set-indexed fractional Brownian motion (sifBm) has been defined by Herbin-Merzbach (2006) for indices that are subsets of a metric measure space. In this paper, the sifBm is proved to statisfy a strenghtened definition of increment stationarity. This new definition for stationarity property allows to get a complete characterization of this process by its fractal properties: The sifBm is the only set-indexed Gaussian process which is self-similar and has stationary increments. Using the fact that the sifBm is the only set-indexed process whose projection on any increasing path is a one-dimensional fractional Brownian motion, the limitation of its definition for a self-similarity parameter 0<H<1/2 is studied, as illustrated by some examples. When the indexing collection is totally ordered, the sifBm can be defined for 0<H<1.