Generic Multifractality in Exponentials of Long Memory Processes

Abstract

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 $ϕ+1/2$, where $ϕ>0$. This generalizes previous studies performed only with $ϕ=0$ (with a truncation at an integral scale), by showing that multifractality holds over a remarkably large range of dimensionless scales for $ϕ>0$. The intermittency multifractal coefficient can be tuned continuously as a function of the deviation $ϕ$ from 1/2 and of another parameter $σ^2$ embodying information on the short-range amplitude of the memory kernel, the ultra-violet cut-off (``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 $ζ(q)$ 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 $ζ(q)$ by different combinations of $ϕ$ and $σ^2$.