Fractional Brownian motion and multifractional Brownian motion of Riemann-Liouville type

The relationship between standard fractional Brownian motion (FBM) and FBM based on the Riemann-Liouville fractional integral (or RL-FBM) is clarified. The absence of stationary property in the increment process of RL-FBM is compensated by a weaker property of local stationarity, and the stationary property for the increments of the large-time asymptotic RL-FBM. Generalization of RL-FBM to the RL-multifractional Brownian motion (RL-MBM) can be carried out by replacing the constant Hölder exponent by a time-dependent function. RL-MBM is shown to satisfy a weaker scaling property known as the local asymptotic self-similarity. This local scaling property can be translated into the small-scale behaviour of the associated scalogram by using the wavelet transform.