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December 2012 A class of multifractal processes constructed using an embedded branching process
Geoffrey Decrouez, Owen Dafydd Jones
Ann. Appl. Probab. 22(6): 2357-2387 (December 2012). DOI: 10.1214/11-AAP834


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.


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Geoffrey Decrouez. Owen Dafydd Jones. "A class of multifractal processes constructed using an embedded branching process." Ann. Appl. Probab. 22 (6) 2357 - 2387, December 2012.


Published: December 2012
First available in Project Euclid: 23 November 2012

zbMATH: 1270.60046
MathSciNet: MR3024971
Digital Object Identifier: 10.1214/11-AAP834

Primary: 60G18
Secondary: 28A80 , 60J85 , 68U20

Keywords: branching process , Brownian motion , multifractal , Self-similar , simulation , time-change

Rights: Copyright © 2012 Institute of Mathematical Statistics


Vol.22 • No. 6 • December 2012
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