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 γ is defined. This new process has two spectral regimes: It behaves like an fBm of parameter H_o for low frequencies |ω|<γ and like an fBm of parameter H_i for high frequencies |ω|≥γ. When H_o=H_i, or for limit cases γ→0 and γ→∞, 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_o and with parameter H_i at low scale.