Dynamic models of long-memory processes driven by Lévy noise. (English) Zbl 1016.60039

Let \({\mathcal D}_t^{\alpha} f(t)\) be the Riemann-Liouville fractional derivative of \(f(t)\). Consider the linear differential operator \(\mathcal L\) with constant coefficients given by \({\mathcal L} y(t) = (A_n{\mathcal D}_t^{\beta_n} + \cdots + A_0{\mathcal D}_t^{\beta_0}) y(t)\) where \(\beta_n>\cdots>\beta_0\) and \(n\geq 1\). The authors analyze fractional differential equations (FDE) of the form \({\mathcal L}X(t) = \dot L(t)\) 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.


60G10 Stationary stochastic processes
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