A New Definition, a Generalisation and an Approximation for a Fractional Derivative with Applications to Stochastic Time Series Modeling.

A brief review of fractional differentiation, fractional integration and differo-integral operators is given based on the properties of the Fourier transform. This is undertaken to provide the reader with a quick-guide and a short background to the fractional calculus and includes a brief discussion on some of the principal characteristics of fractional differointegral operators. The paper then presents a new definition for a fractional differo-integral based on the properties of the sign function and explores some related results. Using the properties of the Dirac delta function, a generalisation is developed in order to quantify the issue as to whether there is an upper bound to the number of definitions for a fractional differo-integral operator that can be developed. Finally, an approximation of a fractional differential is considered and used in the construction of a self-affine stochastic time series model based on the Kolmogorov-Feller equation for the memory function. [ABSTRACT FROM AUTHOR]