${L^p}$-variations for multifractal fractional random walks

TL;DR

This paper extends multifractal random walks to a broader class called multifractal fractional random walks by analyzing their convergence and scaling properties, providing new models and methods for estimating multifractal exponents.

Contribution

It introduces the multifractal fractional random walk as a new extension of MRWs, analyzing its convergence and scaling behavior under certain conditions.

Findings

Established convergence of the new process under specific conditions.

Analyzed the fine-scale scaling structure using empirical structure functions.

Provided methods for inference and confidence intervals for multifractal exponents.

Abstract

A multifractal random walk (MRW) is defined by a Brownian motion subordinated by a class of continuous multifractal random measures $M[0,t], 0\le t\le1$. In this paper we obtain an extension of this process, referred to as multifractal fractional random walk (MFRW), by considering the limit in distribution of a sequence of conditionally Gaussian processes. These conditional processes are defined as integrals with respect to fractional Brownian motion and convergence is seen to hold under certain conditions relating the self-similarity (Hurst) exponent of the fBm to the parameters defining the multifractal random measure $M$. As a result, a larger class of models is obtained, whose fine scale (scaling) structure is then analyzed in terms of the empirical structure functions. Implications for the analysis and inference of multifractal exponents from data, namely, confidence intervals, are also provided.