Generic Multifractality in Exponentials of Long Memory Processes

Alexander I. Saichev, & Didier Sornette

Published July 2006, SCEC Contribution #996

We find that multifractal scaling is a robust property of a large class of continuous stochastic processes, constructed as exponentials of long-memory processes. The long memory is characterized by a power law kernel with tail exponent phi+1/2, where phi > 0. This generalizes previous studies performed only with phi=0 (with a truncation at an integral scale) by showing that multifractality holds over a remarkably large range of dimensionless scales for phi > 0. The intermittency multifractal coefficient can be tuned continuously as a function of the deviation phi from 1/2 and of another parameter sigma(2) embodying information on the short-range amplitude of the memory kernel, the ultraviolet cutoff ("viscous") scale, and the variance of the white-noise innovations. In these processes, both a viscous scale and an integral scale naturally appear, bracketing the "inertial" scaling regime. We exhibit a surprisingly good collapse of the multifractal spectra zeta(q) on a universal scaling function, which enables us to derive high-order multifractal exponents from the small-order values and also obtain a given multifractal spectrum zeta(q) by different combinations of phi and sigma(2). (c) 2006 American Institute of Physics.

Saichev, A. I., & Sornette, D. (2006). Generic Multifractality in Exponentials of Long Memory Processes. Physical Review E, 74(1), 011111. doi: 10.1103/PhysRevE.74.011111.