fbm.msd: Mean square displacement of fractional Brownian motion.
Description
Mean square displacement of fractional Brownian motion.
Usage
fbm.msd(tseq, H)
Arguments
tseq
Length-N vector of timepoints.
H
Hurst parameter (between 0 and 1).
Value
Length-N vector of mean square displacements.
Details
The mean squared displacement (MSD) of a stochastic process \(X_t\) is defined as
$$
\mathrm{\scriptsize MSD}_X(t) = E[(X_t - X_0)^2].
$$
Fractional Brownian motion (fBM) is a continuous Gaussian process with stationary increments, such that its covariance function is entirely defined the MSD, which in this case is \(\textrm{\small MSD}_X(t) = |t|^{2H}\).