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. Generaliza tion of RL-FBM to the RL-multifractional Brownian motion (RL-MBM) can be ca rried out by replacing the constant Holder exponent by a time-dependent fun ction. RL-MBM is shown to satisfy a weaker scaling property known as the lo cal asymptotic self-similarity. This local scaling property can be translat ed into the small-scale behaviour of the associated scalogram by using the wavelet transform.