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Dynamic models of long-memory processes driven by Lévy noise. (English) Zbl 1016.60039
Let 𝒟 t α f(t) be the Riemann-Liouville fractional derivative of f(t). Consider the linear differential operator with constant coefficients given by y(t)=(A n 𝒟 t β n ++A 0 𝒟 t β 0 )y(t) where β n >>β 0 and n1. The authors analyze fractional differential equations (FDE) of the form X(t)=L ˙(t) where 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)= 0 t G n (t-s)dL(s) where G n is the Green function with the Laplace transform g n (p)=(A n p β n ++A 0 p β 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.
MSC:
60G10Stationary processes