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On the $ p$th variation of a class of fractal functions


Authors: Alexander Schied and Zhenyuan Zhang
Journal: Proc. Amer. Math. Soc. 148 (2020), 5399-5412
MSC (2010): Primary 60H05, 28A80, 26A45, 60E05
DOI: https://doi.org/10.1090/proc/15171
Published electronically: September 11, 2020
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Abstract: The concept of the $ p$th variation of a continuous function $ f$ along a refining sequence of partitions is the key to a pathwise Itô integration theory with integrator $ f$. Here, we analyze the $ p$th variation of a class of fractal functions, containing both the Takagi-van der Waerden and Weierstraß functions. We use a probabilistic argument to show that these functions have linear $ p$th variation for a parameter $ p\ge 1$, which can be interpreted as the reciprocal Hurst parameter of the function. It is shown, moreover, that if functions are constructed from (a skewed version of) the tent map, then the slope of the $ p$th variation can be computed from the $ p$th moment of a (non-symmetric) infinite Bernoulli convolution. Finally, we provide a recursive formula of these moments and use it to discuss the existence and non-existence of a signed version of the $ p$th variation, which occurs in pathwise Itô calculus when $ p\ge 3$ is an odd integer.


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Additional Information

Alexander Schied
Affiliation: Department of Statistics and Actuarial Science, University of Waterloo, Canada
MR Author ID: 604157
ORCID: 0000-0003-0296-7701
Email: aschied@uwaterloo.ca

Zhenyuan Zhang
Affiliation: Department of Statistics and Actuarial Science, University of Waterloo, Canada
Email: z569zhan@uwaterloo.ca

DOI: https://doi.org/10.1090/proc/15171
Keywords: $p$th variation, Weierstra{\ss} function, Takagi-van der Waerden functions, pathwise It\^o calculus, (non-symmetric) infinite Bernoulli convolution and its moments
Received by editor(s): July 2, 2019
Received by editor(s) in revised form: April 26, 2020
Published electronically: September 11, 2020
Additional Notes: The authors gratefully acknowledge financial support from the Natural Sciences and Engineering Research Council of Canada through grant RGPIN-2017-04054
Communicated by: Zhen-Qing Chen
Article copyright: © Copyright 2020 American Mathematical Society