Stochastic Analysis and Related Topics by Fabrice Baudoin & Jonathon Peterson

Author:Fabrice Baudoin & Jonathon Peterson
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

where B HK is a fBm with Hurst parameter HK, the process Y K is given by

(1.4)

with W = {W y , y ≥ 0} a standard Brownian motion independent of B H, K , and C 1, C 2 are constants given by and . The process Y K has trajectories which are infinitely differentiable on (0, ∞) and Hölder continuous of order HK −ε in any interval [0, T] for any ε > 0. In particular, this leads to a decomposition in law of the process U with covariance (1.2) as the sum of a fractional Brownian motion with Hurst parameter plus a regular process.

The classical one-dimensional space-time white noise can also be considered as an extension of the covariance (1.2) if we take β = 1. In this case the covariance corresponds, up to a constant, to that of a bifractional Brownian motion with parameters .

The case where the noise term is a fractional Brownian motion with Hurst parameter in time and a spatial covariance given by the Riesz kernel, that is,

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