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 $\varphi +1/2$, where $\varphi >0$. This generalizes previous studies performed only with $\varphi =0$ (with a truncation at an integral scale), by showing that multifractality holds over a remarkably large range of dimensionless scales for $\varphi >0$. The intermittency multifractal coefficient can be tuned continuously as a function of the deviation $\varphi$ from 1/2 and of another parameter ${\sigma }^2$ 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 $\zeta (q)$ 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 $\varphi$ and ${\sigma }^2$.

10 pages + 9 figures

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