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

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

2008-06-17

arxiv.org/abs/0806.2731v1

Annals of Applied Probability 2008, Vol. 18, No. 3, 1138-1163

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/07-AAP483 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Inst

Scientific paper

10.1214/07-AAP483

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.

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