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Zbl 1083.60029
Lim, S.C.; Muniandy, S.V.
Generalized Ornstein-Uhlenbeck processes and associated self-similar processes.
(English)
[J] J. Phys. A, Math. Gen. 36, No. 14, 3961-3982 (2003). ISSN 0305-4470

Fractional Brownian motion (FBM) $X=\{X(t), t \ge 0\}$ is the unique self-similar Gaussian process with stationary increments. Through Lamperti's transformation, it corresponds to the stationary Gaussian process $Y_1(t) = e^{- c H t} X(e^{ct})$, which is called the Ornstein-Uhlenbeck process associated to FBM. Besides $Y_1$, the authors also consider other two stationary Gaussian processes $Y_2$ and $Y_3$, where $Y_2$ has its covariance function given by $$C_2(\tau) = \langle Y_2(t+\tau) Y_2(t)\rangle = A \, e^{- a \vert \tau\vert ^\alpha}$$ and $Y_3$ is specified by a fractional Langevin equation which can be represented as $$Y_3(t) = c (a, \beta) \int_{-\infty}^t {{(t-u)^{\beta - 1} e^{- a (t-u)}}\over {\Gamma(\beta)}}\, \eta(u) du,$$ where $c (a, \beta) > 0$ is a normalization constant and $\eta$ is the white noise. Since the covariance function of $Y_3$ is related to the modified Bessel function, the authors call $Y_3$ the $K$-Bessel process. \par The authors compare the asymptotic behavior of the covariance functions and the spectral densities of $Y_1$, $Y_2$ and $Y_3$ and show that they have many common properties such as the covariance functions have similar local structures and their spectral density functions have similar asymptotic properties at large frequency. They argue that these stationary Gaussian processes can be regarded as the local stationary representations of FBM. They also consider the self-similar Gaussian processes $X_2$ and $X_3$ obtained from $Y_2$ and $Y_3$ via the (inverse) Lamperti transformation. They show that, even though $X_2$ and $X_3$ do not have stationary increments, the variances of their increments behave locally like that of FBM and the problem of long-range dependence can be studied. They give simulation of the sample paths of these Gaussian processes based on numerical Karhunen-Loève expansion.
[Yimin Xiao (East Lansing)]
MSC 2000:
*60G15 Gaussian processes
60G18 Self-similar processes
60G10 Stationary processes

Keywords: fractional Brownian motion, Lamperti's transformation; fractional Langevin equation; local stationarity

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