Fractional Brownian motion approximation based on fractional integration of a white noise (Q5939056)

Let \(B_H\) be a fractional Brownian motion, where the self-similarity parameter \(H\) satisfies \(0<H<1\). Define the fractional Gaussian noise by the formula NEWLINE\[NEWLINEB_H'(t,\delta)= {1\over\delta} \bigl(B(t+ \delta)-B_H(t) \bigr). NEWLINE\]NEWLINE Here \(\delta >0\) is a fixed parameter, describing the smallest time interval relevant to the problem. The authors give a method to simulate fractional Gaussian noise. The method starts with the Gaussian white noise \(X(t)\). Then consider its Fourier transform \(\widehat X(f)\), multiply it by \(f^\nu\) and use the inverse Fourier transform to obtain an approximation to the fractional Gaussian noise and use these differences to obtain an approximation to fractional Brownian motion. The authors end the paper by giving some results based on simulation.