In all articles about fBm this formula of variance is given as the definition:
$$ Cov\big[B_H(t),B_H(s)\big] = \frac{1}{2}\big(|t|^{2H}+|s|^{2H}-|t-s|^{2H}\big).$$
However, nowhere I could find the derivation of it from the original fBm formula:
$$B_H(t) = \frac{1}{\Gamma(H+1/2)}\int_{0}^{t}(t-s)^{H-1/2}dB(s).$$
Could you help me to derive it or give a link where it can be found?
The fBm with the Hurst parameter $H\in (0,1)$, $B_{t}^{H}$ for $t\in \mathbb{% R}$, is a zero mean Gaussian process with covariance \begin{equation*} \mathbb{E}(B_{t}^{H}B_{s}^{H})=R_{H}(s,t)=\frac{1}{2}\left( |t|^{2H}+|s|^{2H}-|t-s|^{2H}\right) \,. \label{cov} \end{equation*}
For $t>0$, it has the following integral representation: \begin{equation*} B_{t}^{H}=\frac{1}{c_{H}}\left\{ \int_{-\infty }^{0}\left[ (t-u)^{H-1/2}-(-u)^{H-1/2}\right] dW_{u}+\int_{0}^{t}(t-u)^{H-1/2}dW_{u}% \right\} , \label{mand} \end{equation*} where $W_{t}$ is a standard Brownian motion, and $c_{H}=\Gamma(H+1/2)$.
For details, please see Robinso or Davidson.