How rich is the class of multifractional Brownian motions?

Abstract

The multifractional Brownian motion (MBM) processes are locally self-similar Gaussian processes. They extend the classical fractional Brownian motion processes B H = { B H ( t ) } t ∈ R by allowing their self-similarity parameter H ∈ ( 0 , 1 ) to depend on time. Two types of MBM processes were introduced independently by Peltier and Lévy-Vehel [Multifractional Brownian motion: definition and preliminary results, Technical Report 2645, Institut National de Recherche en Informatique et an Automatique, INRIA, Le Chesnay, France, 1995] and Benassi, Jaffard, Roux [Elliptic Gaussian random processes, Rev. Mat. Iber. 13(1) (1997) 19–90] by using time-domain and frequency-domain integral representations of the fractional Brownian motion, respectively. Their correspondence was studied by Cohen [From self-similarity to local self-similarity: the estimation problem, in: M. Dekking, J.L. Véhel, E. Lutton, C. Tricot (Eds.), Fractals: Theory and Applications in Engineering, Springer, Berlin, 1999]. Contrary to what has been stated in the literature, we show that these two types of processes have different correlation structures when the function H ( t ) is non-constant. We focus on a class of MBM processes parameterized by ( a + , a - ) ∈ R 2 , which contains the previously introduced two types of processes as special cases. We establish the connection between their time- and frequency-domain integral representations and obtain explicit expressions for their covariances. We show, that there are non-constant functions H ( t ) for which the correlation structure of the MBM processes depends non-trivially on the value of ( a + , a - ) and hence, even for a given function H ( t ) , there are an infinite number of MBM processes with essentially different distributions.