For a standard Brownian motion $(B_t)_{t \geq 0}$, we have $$B_t \sim\mathcal{N}(0, t),\quad B_t-B_s\sim\mathcal{N}(0, t-s)$$
Then, what are the distributions of increments for a fractional Brownian motion $(B_t^H)_{t \geq 0}$?
I think there should be a $t^{2H}$ term involved since the autocovariance function is $$\mathbb{E}[B^H_tB^H_s] = \frac{1}{2}\left(|t|^{2H}+|s|^{2H}-|t-s|^{2H}\right)$$
Best Answer
The fractional Brownian motion with Hurst parameter $H > 0$ is a centered Gaussian process and hence its increments are centered Gaussians. To completely characterise the distribution of $B_t^H - B_s^H$ it thus suffices to find its variance. This is a simple calculation using the known form of the covariance of $(B_t^H)$.
$$\mathbb{E}[(B_t^H - B_s^H)^2 ] = |t|^H + |s|^H - \left(|t|^{2H}+|s|^{2H}-|t-s|^{2H}\right) = |t-s|^{2H}$$