Definition#

A fractional Brownian motion (fBm) is a continuous Gaussian process $W^H$ with mean $0$ and covariance function defined as follows:

$$ \mathbb{E}[W^H_tW^H_s] := \frac{1}{2}\Big(|t|^{2H} + |s|^{2H} - |t - s|^{2H}\Big) $$

where $H \in \left]0, 1\right[$ is known as the Hust index/parameter of the given fBm.

Properties#

Mandelbrot and Van Ness representation:

$$ W^H_t := C_H\Big(\int_0^t (t - s)^{H - \frac{1}{2}}\mathrm{d}W_s + \int_{- \infty}^0 \Big[(t - s)^{H - \frac{1}{2}} - (-s)^{H - \frac{1}{2}}\Big]\mathrm{d}W_s\Big) $$