Correlation structure of fractional Pearson diffusions

Authors:
Nikolai N. Leonenko;Mark M. Meerschaert;Alla Sikorskii
Affiliations:
Cardiff School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF24 4YH, UK;Department of Statistics and Probability, Michigan State University, East Lansing, MI 48824, USA;Department of Statistics and Probability, Michigan State University, East Lansing, MI 48824, USA
Venue:
Computers & Mathematics with Applications
Year:
2013

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Abstract

The stochastic solution to a diffusion equations with polynomial coefficients is called a Pearson diffusion. If the first time derivative is replaced by a Caputo fractional derivative of order less than one, the stochastic solution is called a fractional Pearson diffusion. This paper develops an explicit formula for the covariance function of a fractional Pearson diffusion in steady state, in terms of Mittag-Leffler functions. That formula shows that fractional Pearson diffusions are long-range dependent, with a correlation that falls off like a power law, whose exponent equals the order of the fractional derivative.