Fractional Brownian Motion

Similar to a Wiener Process $B_t$, differing in the covariance: $\mathbb{E}[B(t)B(s)=\frac{1}{2}\left(|t{|}^{2H}+|s{|}^{2H}-|t-s{|}^{2H}\right)]$, where $H\in (0,1)$, called the Hurst index, characterizes the correlations of this Wiener Process:

• $H=\frac{1}{2}$ is the usual Wiener Process
• $H>\frac{1}{2}$ means increments are positively correlated.
• $H<\frac{1}{2}$ means the increments are negatively correlated.