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

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- by Alexander Schied and Zhenyuan Zhang
- Proc. Amer. Math. Soc.
**148**(2020), 5399-5412 - 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.## References

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## Bibliographic Information

**Alexander Schied**- Affiliation: Department of Statistics and Actuarial Science, University of Waterloo, Canada
- Email: aschied@uwaterloo.ca
**Zhenyuan Zhang**- Affiliation: Department of Statistics and Actuarial Science, University of Waterloo, Canada
- Email: z569zhan@uwaterloo.ca
- 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
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**148**(2020), 5399-5412 - MSC (2010): Primary 60H05, 28A80, 26A45, 60E05
- DOI: https://doi.org/10.1090/proc/15171
- MathSciNet review: 4163851