Affine stationary processes with applications to fractional Brownian motion

In our previous work, we introduced a new class of nonstationary stochastic processes whose spectral representation is associated with the wavelet transforms and established a mathematical framework for the analysis of such processes [1]. We refer to these processes as affine stationary processes. These processes are indexed by the affine group, or ax + b group, which can be though of as a group of shifts and scalings. Affine stationary processes are nonstationary in the classical sense. However, their second order statistical properties are invariant under the affine group composition law. In this paper, we show that any physically realizable affine stationary process is a wavelet transform of the white noise process. As a result, we derive a spectral decomposition of the affine stationary processes using wavelet transform. Additionally, we apply our results to the fractional Brownian motion (fBm). We show that fBm is an affine stationary process and the filter associated with the fBm is a continuous time analyzing wavelet. Finally, we apply our results to choose an optimal wavelet filter in the development of a spectral representation of fBm via wavelet transforms.