Non-Singular Kernels for Modelling Power Law Type Long Memory Behaviours and Beyond

Abstract

To tackle several limitations recently highlighted in the field of fractional differentiation and fractional models, some authors have proposed new kernels for the definition of fractional integration/differentiation operators. These kernels still have some limitations, however. One goal of this paper is thus to propose other kernels with a power law type behavior that overcome these limitations. All these kernels are nonsingular and have a limited memory. They are then used to define a class of models adapted to capture input-output power law type long memory behaviors. The stability of this class of model is investigated. Finally, the paper shows that the fractional pseudo state space description, a fractional model widely used in the literature, is a special case of Volterra equations, equations introduced nearly a century ago. Volterra equations, whose memory length can be explicitly controlled, can thus be viewed as a serious alternative to fractional pseudo state space descriptions for power law type long memory behavior modeling, as fractional pseudo state space descriptions are known to exhibit serious drawbacks also discussed in the paper.