The Annals of Applied Probability

A class of multifractal processes constructed using an embedded branching process

Geoffrey Decrouez and Owen Dafydd Jones

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We present a new class of multifractal process on $\mathbb{R}$, constructed using an embedded branching process. The construction makes use of known results on multitype branching random walks, and along the way constructs cascade measures on the boundaries of multitype Galton–Watson trees. Our class of processes includes Brownian motion subjected to a continuous multifractal time-change.

In addition, if we observe our process at a fixed spatial resolution, then we can obtain a finite Markov representation of it, which we can use for on-line simulation. That is, given only the Markov representation at step $n$, we can generate step $n+1$ in $O(\log n)$ operations. Detailed pseudo-code for this algorithm is provided.

Article information

Ann. Appl. Probab., Volume 22, Number 6 (2012), 2357-2387.

First available in Project Euclid: 23 November 2012

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Zentralblatt MATH identifier

Primary: 60G18: Self-similar processes
Secondary: 28A80: Fractals [See also 37Fxx] 60J85: Applications of branching processes [See also 92Dxx] 68U20: Simulation [See also 65Cxx]

Self-similar multifractal branching process Brownian motion time-change simulation


Decrouez, Geoffrey; Jones, Owen Dafydd. A class of multifractal processes constructed using an embedded branching process. Ann. Appl. Probab. 22 (2012), no. 6, 2357--2387. doi:10.1214/11-AAP834.

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