Piecewise fractional Brownian motion

Abstract

Starting from fractional Brownian motion (fBm) of unique parameter H, a piecewise fractional Brownian motion (pfBm) of parameters H/sub o/, H/sub i/, and /spl gamma/ is defined. This new process has two spectral regimes: It behaves like an fBm of parameter H/sub o/ for low frequencies |/spl omega/|</spl gamma/ and like an fBm of parameter H/sub i/ for high frequencies |/spl omega/|/spl ges//spl gamma/. When H/sub o/=H/sub i/, or for limit cases /spl gamma//spl rarr/0 and /spl gamma//spl rarr//spl infin/, 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/sub o/ and with parameter H/sub i/ at low scale.