Fractals and Stochastic Calculus

Standard Deviation of Fractional Brownian Motion

Let $B_{H}(t)$ be the zero-mean Fractional Brownian Motion process of Hurst parameter $H (0 \le H \le 1)$, such a process is defined as having the following covariance structure:

$E[B_{H}(t)B_{H}(s)]=\frac{1}{2}(|t|^{2H}+|s|^{2H}-|t-s|^{2H})$ (1)

Then(by taking s=t in (1))

$Var(B_{H}(t))=E[(B_{H}(t)-E(B_{H}(t)))^2]=E[B_{H}^{2}(t)]=|t|^{2H}$

Which gives the standard deviation as $|t|^H$ (and for the Wiener Brownian Motion, we indeed get a standard deviation of $\sqrt{t}$)