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Dynamic models of long-memory processes driven by Lévy noise. (English) Zbl 1016.60039
Let ${\Cal D}_t^{\alpha} f(t)$ be the Riemann-Liouville fractional derivative of $f(t)$. Consider the linear differential operator $\Cal L$ with constant coefficients given by ${\Cal L} y(t) = (A_n{\Cal D}_t^{\beta_n} + \cdots + A_0{\Cal D}_t^{\beta_0}) y(t)$ where $\beta_n>\cdots>\beta_0$ and $n\ge 1$. The authors analyze fractional differential equations (FDE) of the form ${\Cal L}X(t) = \dot L(t)$ where $\dot L$ is the derivative of a Lévy process in the distribution sense. The Green function solution of the FDE is of the form $X_n(t) = \int_0^t G_n(t-s) dL(s)$ where $G_n$ is the Green function with the Laplace transform $g_n(p) = (A_np^{\beta_n} + \cdots + A_0p^{\beta_0})^{-1}$. Some exact results on the Green functions, covariance functions, spectra, and higher-order spectra of particular forms of FDE are obtained. The FDE can be applied to stochastic volatility of asset prices and to macroeconomics.

60G10Stationary processes
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