Multifractal Random Walks as Fractional Wiener Integrals

Abstract

Multifractal random walks are defined as integrals of infinitely divisible stationary multifractal cascades with respect to fractional Brownian motion. Their key properties are studied, such as finiteness of moments and scaling, with respect to the chosen values of the self-similarity and infinite divisibility parameters. The range of these parameters is larger than that considered previously in the literature, and the cases of both exact and nonexact scale invariance are considered. Special attention is paid to various types of definitions of multifractal random walks. The resulting random walks are of interest in modeling multifractal processes whose marginals exhibit stationarity and symmetry.