Piecewise fractional Brownian motion

Starting from fractional Brownian motion (fBm) of unique parameter H, a piecewise fractional Brownian motion (pfBm) of parameters H-o H-i, and gamma is defined. This new process has two spectral regimes: It behaves like an fBm of parameter H-o for low frequencies \omega\ < gamma and like an fBm of parameter H-i for high frequencies \omega\ greater than or equal to gamma. When H-o = H-i, or for limit cases gamma --> 0 and gamma --> infinity, pfBm becomes classical fBm. It is shown that pfBm is a continuous, Gaussian, and nonstationary process having continuous, Gaussian, and stationary increments, namely, piecewise fractional Gaussian noises. The asymptotic self-similarity of pfBm is shown according to the considered regime: At large scale, the process is self-similar with parameter H. and with parameter Hi at low scale.