📂Probability Theory

Fractal Brownian Motion

Definition

$E \left( X_{t} \right) = 0$ given $X_{t}$ is a Gaussian process and let’s denote it by $H \in (0, 1)$. The Fractional Brownian motion can be defined in the following two ways.

Definition through Covariance ¹

The covariance at time point $t, s$ of $X_{t}$, if it follows the expression below, is referred to as Fractional Brownian Motion. $$\operatorname{Cov} \left( X_{t}, X_{s} \right) = {{ 1 } \over { 2 }} \left( t^{2H} + s^{2H} - \left| t-s \right|^{2H} \right)$$

Definition through Conditions ²

$X_{t}$ is considered as Fractional Brownian Motion if it satisfies the following two conditions.

• (i): It has stationary increments.
• (ii): It is Hurst index $H$ self-similar with respect to $H$.

Explanation

The term Fractional is thought to be more appropriately derived from Fractal rather than Fraction considering the self-similarity mentioned in the definition, thus it is rationalized to Fractional Brownian Motion.

When $H = 1/2$, it is exactly a Brownian Motion. In other words, FBM is a perfect generalization of standard BM.