Lamperti transform and a series decomposition of fractional Brownian motion

Anastassia Baxevani and Krzysztof Podgórski

Abstract:: The Lamperti transformation of a self-similar process is a strictly stationary process. In particular, the fractional Brownian motion transforms to the second order stationary Gaussian process. This process is represented as a series of independent processes. The terms of this series are Ornstein-Uhlenbeck processes if H < 1/2, and linear combinations of two dependent Ornstein-Uhlenbeck processes whose two dimensional structure is Markovian if H > 1/2. From the representation effective approximations of the process are derived. The corresponding results for the fractional Brownian motion are obtained by applying the inverse Lamperti transformation. Implications for simulating the fractional Brownian motion are discussed

Key words:: spectral density, covariance function, stationary Gaussian processes, long-range dependence