circFBM: Simulation of a fractional Brownian motion by using the circulant matrix method
Description
Generates a discretized sample path of a fBm, $B_H=(B_H(0),...,B_H(n-1/n))$, at times $0,...,(n-1)/n$ with Hurst parameter H in (0,1) by using the circulant matrix method. A fBm with scaling coefficient $C>0$ and discretized at times 0,...,n-1 is obtained by the operation: $n^H * C * B_H$.
Usage
circFBM(n, H, plotfBm = FALSE)
Arguments
n
sample size
H
Hurst parameter
plotfBm
possible plot of the generated sample path
Value
Returns a vector of length n.
Details
The circulant matrix method consists in embedding the covariance matrix of the increments of the fractional Brownian motion (which is a Toeplitz matrix since the increments are stationary) in a matrix, say $M$, whose size is a power of 2 greater than n. One then uses general results on circulant matrices to compute easily and very quickly the eigenvalues of $M$. Note that the simulation fails if the procedure does not find a matrix $M$ such that all its eigenvalues are positive.
References
J.-F. Coeurjolly (2001) Simulation and identification of the fractional Brownian motion: a bibliographic and comparative study. Journal of Statistical Software, Vol. 5.
A.T.A. Wood and G. Chan (1994) Simulation of stationary Gaussian processes in $[0,1]^d$. Journal of computational and graphical statistics, Vol. 3 (4), p.409--432.