Fractional differences, derivatives and fractal time series. (Q2773688)

This article starts with a brief review of three approaches in modelling fractal time series. Some of the difficulties in modelling stochastic processes with long time memory are pointed out.NEWLINENEWLINE Next, the authors investigate fractional differences between discrete time steps and define the fractional derivatives in terms of the continuum limit of the step length. This approach is further extended to discrete stochastic processes, where the fractional difference is related to the long-term memory of the system's response to white noise.NEWLINENEWLINE Further a connection is derived between the Riemann-Liouville fractional derivative of the probability density in space with the fractional/diffusion equation. This is done by finding the relationship between the Riemann-Liouville fractional derivative and the limit of fractional-difference operators.NEWLINENEWLINE Finally, it is interesting to note that one of the applications of the fractional-difference approach is to model both the long and short time properties of a time series.NEWLINENEWLINEFor the entire collection see [Zbl 0998.26002].