SPECTRAL DENSITY OF A FRACTIONAL BROWNIAN PROCESS
E. L. Kuleshov, B. N. Grudin
Far Eastern Federal University, ul. Sukhanov 8, Vladivostok, 690950 Russia
kuleshov@lemoi.phys.dvgu.ru
Keywords: fractional Brownian process, correlation function, spectral density, periodogram
Abstract
A model of a fractional Brownian process is defined by its structural function with a Hurst exponent α ∈ (0, 1). It is proved that the spectral density of this process exists and coincides with the known power-law relationship only for values of the exponent α ∈ (0, 1/2]. In the range of α ∈ (1/2, 1), the spectral density does not exist and the periodogram estimate of the exponent has a constant value equal to 1/2. The theoretical results were verified by modeling trajectories of the process, calculating periodograms, and estimating the spectral density exponent.
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