A Multifractal Walk down Wall Street 1 The very fact that the Petersburg Problem has not yielded a unique and generally acceptable solution to more than 200 years of attack by some of the world’s great intellects suggests, indeed, that the growth-stock problem offers no hope of a satisfactory solution. David Durand.
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A class of multifractal processes constructed using an embedded branching process
Geoffrey Decrouez and Owen Dafydd Jones
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Abstract
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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Source
Ann. Appl. Probab., Volume 22, Number 6 (2012), 2357-2387.
Dates
First available in Project Euclid: 23 November 2012
Permanent link to this document
https://projecteuclid.org/euclid.aoap/1353695956
Digital Object Identifier
doi:10.1214/11-AAP834
Mathematical Reviews number (MathSciNet)
MR3024971
Zentralblatt MATH identifier
1270.60046
Subjects
Primary: 60G18: Self-similar processes Secondary: 28A80: Fractals [See also 37Fxx]60J85: Applications of branching processes [See also 92Dxx]68U20: Simulation [See also 65Cxx]
Keywords
Self-similarmultifractalbranching processBrownian motiontime-changesimulation Citation
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. https://projecteuclid.org/euclid.aoap/1353695956
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