Fractional Brownian motion with variable Hurst parameter: definition and properties
The author introduces a class of Gaussian processes generalizing the fractional Brownian motion with Hurst index \(H\in (1/2,1).\) Any measurable function attaining values in the interval \((1/2,1)\) can be chosen as a variable Hurst parameter for this new class. These processes allow modeling of phenomena where long-range dependence is present and self-similarity changes as the phenomenon evolves. Sample path properties of these processes are investigated. Fokker-Plank-type equations for such processes are derived. It is found that the regularity properties of the chosen Hurst function directly correspond to the regularity of the sample paths of the process. Reviewer's comments: Such processes are also called locally self-similar. \textit{P. Gonçalves} and \textit{P. Flandrin} [in: Progress in wavelet analysis and applications. Proceedings of the 3rd international conference on wavelets and applications, Toulouse, France, 1992. Gif-sur-Yvette: Editions Frontières, 271--276 (1993; Zbl 0878.94024)] proposed such processes. \textit{Y.-Z. Wang} et al. [J. Stat. Plann. Inference 99, No. 1, 91--110 (2001; Zbl 0989.62045)] studied estimations of the self-similarity index for locally self-similar processes (cf. [\textit{B. L. S. Prakasa Rao}, Statistical inference for fractional diffusion processes. Wiley Series in Probability and Statistics. Chichester: John Wiley \& Sons (2010; Zbl 1211.62143)]).
scientific article