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A probabilistic approach to the Φ-variation of classical fractal functions with critical roughness

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  • Han, Xiyue
  • Schied, Alexander
  • Zhang, Zhenyuan

Abstract

We consider Weierstraß and Takagi–van der Waerden functions with critical degree of roughness. In this case, the functions have vanishing pth variation for all p>1 but are also nowhere differentiable and hence not of bounded variation either. We resolve this apparent puzzle by showing that these functions have finite, nonzero, and linear Wiener–Young Φ-variation along the sequence of b-adic partitions, where Φ(x)=x∕−logx. For the Weierstraß functions, our proof is based on the martingale central limit theorem (CLT). For the Takagi–van der Waerden functions, we use the CLT for Markov chains if a certain parameter b is odd, and the standard CLT for b even.

Suggested Citation

  • Han, Xiyue & Schied, Alexander & Zhang, Zhenyuan, 2021. "A probabilistic approach to the Φ-variation of classical fractal functions with critical roughness," Statistics & Probability Letters, Elsevier, vol. 168(C).
    • Handle: RePEc:eee:stapro:v:168:y:2021:i:c:s0167715220302236
    DOI: 10.1016/j.spl.2020.108920
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    References listed on IDEAS

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    1. Hans Follmer & Alexander Schied, 2013. "Probabilistic aspects of finance," Papers 1309.7759, arXiv.org.
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    Cited by:

    1. Xiyue Han & Alexander Schied & Zhenyuan Zhang, 2022. "A Limit Theorem for Bernoulli Convolutions and the $$\Phi $$ Φ -Variation of Functions in the Takagi Class," Journal of Theoretical Probability, Springer, vol. 35(4), pages 2853-2878, December.
    2. Xiyue Han & Alexander Schied, 2021. "The roughness exponent and its model-free estimation," Papers 2111.10301, arXiv.org, revised Aug 2023.

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