FBM

Fractional Brownian motion (fBm) is a centered Gaussian process B_H(t), t ≥ 0, characterized by its mean zero and a specific covariance structure: E[B_H(s) B_H(t)] = 1/2 (t^{2H} + s^{2H} − |t − s|^{2H}), where H ∈ (0,1) is called the Hurst exponent. When H = 1/2, fBm reduces to standard Brownian motion. For H ≠ 1/2, the process exhibits dependent increments: long-range dependence for H > 1/2 and anti-persistence for H < 1/2.

Key properties include self-similarity with exponent H, meaning the distribution of {B_H(ct)} is the same as {c^H B_H(t)} up to time scaling. fBm has stationary increments and is a continuous, Gaussian process. The variance of increments satisfies E[(B_H(t+τ) − B_H(t))^2] ∝ |τ|^{2H}. Sample paths are almost surely Hölder continuous of any order α < H and are nowhere differentiable for H in (0,1).

Construction and analysis of fBm use several integral representations, notably the Mandelbrot–Van Ness and Molchan–Golosov forms, which express fBm as an integral with respect to standard Brownian motion. It is not a semimartingale unless H = 1/2, which affects the applicability of classical Itô calculus and motivates alternative stochastic integration frameworks, such as Malliavin calculus or pathwise integrals.

Applications of fBm appear in fields that require modeling of self-similarity and long-range dependence, including finance, hydrology, network traffic, and telecommunications. Estimation of the Hurst exponent from data is a common task, and there are variants such as multifractional Brownian motion, where H can vary over time. The standard Brownian motion is the special case H = 1/2.