Wavelet-based synthesis of the multifractional Brownian motion

The multifractional Brownian motion (mBm) is introduced as a natural extension of traditional fractional Brownian motion (fBm). The selling point of mBm is that its Holder regularity is allowed to vary from point to point, such that makes it a promising model for those stochastic processes whose regularity evolves in time. A wavelet-based algorithm to synthesize a realization of mBm is proposed in this work. The desired local regularity of the multifractional process is obtained by controlling the weights of the wavelet expansion of the Gaussian white noise. This approach is not only time saving, also appropriate for generating the multifractional process that is non-Gaussian and autocovariance function unknown in advance. The validity is verified by numerical experiments and real-world data.