Theory of fractional Brownian motion (fBm) is exclusively developed by many researchers and used for modeling long-range dependence when studying processes in computer networks, in financial markets as well as in hydromechanics, climatology, and hydrography.

The fBm  where  is a Gaussian process with stationary increments and has the so-called Hurst parameter  which characterizes self-similarity of distributions and roughness of paths. However, the stationarity of increments of fBm restricts substantially its applicability for modeling processes with long memory. In particular, it does not allow us to model processes whose regularity of paths and “memory depth” change in time. A generalization of the fBm is the multifractional Brownian motion (mBm), denoted by  where the constant Hurst parameter Hin  is substituted by a time-dependent Hölder continuous Hurst function  taking its values in  As such, mBms are useful as stochastic models for phenomena that exhibit non-stationarity, for example, risky asset in financial market, traffic in modern telecommunication networks or signal processing.

fractional Brownian motion, multifractional Brownian motion, stochastic differential equation, S-transform, white noise theory, Wick-Ito integral, Ito formula, asset pricing, Black-Scholes equation.