Modeling of locally self-similar processes using multifractional Brownian motion of Riemann-Liouville type.

Abstract

Fractional Brownian motion (FBM) is widely used in the modeling of phenomena with power spectral density of power-law type. However, FBM has its limitation since it can only describe phenomena with monofractal structure or a uniform degree of irregularity characterized by the constant Holder exponent. For more realistic modeling, it is necessary to take into consideration the local variation of irregularity, with the Holder exponent allowed to vary with time (or space). One way to achieve such a generalization is to extend the standard FBM to multifractional Brownian motion (MBM) indexed by a Holder exponent that is a function of time. This paper proposes an alternative generalization to MBM based on the FBM defined by the Riemann-Liouville type of fractional integral. The local properties of the Riemann-Liouville MBM (RLMBM) are studied and they are found to be similar to that of the standard MBM. A numerical scheme to simulate the locally self-similar sample paths of the RLMBM for various types of time-varying Holder exponents is given. The local scaling exponents are estimated based on the local growth of the variance and the wavelet scalogram methods. Finally, an example of the possible applications of RLMBM in the modeling of multifractal time series is illustrated.