Fractional Brownian motion

Remarks

For $H = 1/2$ the covariance is $\frac{1}{2}(s + t - \abs{t - s}) = \min(s, t)$, so $B^{1/2}$ is a Wiener process. Clause 2 gives $B^H_0 = 0$ and $E[(B^H_t - B^H_s)^2] = \abs{t - s}^{2H}$, so the increments are stationary and the process is self-similar with index $H$. The value $H = 1$ is excluded: the formula then gives covariance $st$, realised by the degenerate process $t \xi$ with $\xi$ standard normal, which has no randomness in its shape. Disjoint increments are positively correlated for $H > 1/2$ and negatively correlated for $H < 1/2$. Almost every path is Hölder continuous of every order less than $H$, and for $H \ne 1/2$ the process is not a semimartingale.