A fractional Brownian field indexed by L2 and a varying Hurst parameter

Alexandre Richard

Stochastic Processes and their Applications, 2015, vol. 125, issue 4, 1394-1425

Abstract: Using structures of Abstract Wiener Spaces, we define a fractional Brownian field indexed by a product space (0,1/2]×L2(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 motions and multiparameter fractional Brownian motions, 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.

Keywords: (multi)fractional Brownian motion; Gaussian fields; Gaussian measures; Abstract Wiener Spaces; Multiparameter and set-indexed processes; Sample paths properties (search for similar items in EconPapers)
Date: 2015
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Citations: View citations in EconPapers (4)

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DOI: 10.1016/j.spa.2014.11.003

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