# A fractional Brownian field indexed by $L^2$ and a varying Hurst parameter

1 TOSCA - TO Simulate and CAlibrate stochastic models
CRISAM - Inria Sophia Antipolis - Méditerranée , IECL - Institut Élie Cartan de Lorraine : UMR7502
Abstract : Using structures of Abstract Wiener Spaces and their reproducing kernel Hilbert spaces, we define a fractional Brownian field indexed by a product space $(0,1/2] \times L^2(T,m)$, $(T,m)$ a separable measure space, where the first coordinate corresponds to the Hurst parameter of fractional Brownian motion. This field encompasses a large class of existing fractional Brownian processes, such as Lévy fractional Brownian motion and multiparameter fractional Brownian motion, and provides a setup for new ones. We prove that it has satisfactory incremental variance in both coordinates and derive certain continuity and Hölder regularity properties in relation with metric entropy. Also, a sharp estimate of the small ball probabilities is provided, generalizing a result on Lévy fractional Brownian motion. Then, we apply these general results to multiparameter and set-indexed processes, proving the existence of processes with prescribed local Hölder regularity on general indexing collections.
Type de document :
Article dans une revue
Stochastic Processes and their Applications, Elsevier, 2015, 125
Domaine :

https://hal.inria.fr/hal-00922028
Contributeur : Alexandre Richard <>
Soumis le : lundi 23 décembre 2013 - 11:06:57
Dernière modification le : vendredi 12 janvier 2018 - 01:53:58

### Identifiants

• HAL Id : hal-00922028, version 1
• ARXIV : 1312.6069

### Citation

Alexandre Richard. A fractional Brownian field indexed by $L^2$ and a varying Hurst parameter. Stochastic Processes and their Applications, Elsevier, 2015, 125. 〈hal-00922028〉

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