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

2019/09/11 by Alexander Schied, Zhenyuan Zhang, Schied, Alexander +1
Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Approximation and Integration #Mathematical Dynamics and Fractals #Probability (math.PR) #advanced mathematical theories #math.CA #math.PR

paper · pdf · doi:10.48550/arxiv.1909.05239

openalex publication_date 2019/09/11 · arxiv created 2020/04/27 · arxiv updated 2020/04/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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≥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≥3 is an odd integer.

Related

• A limit theorem for Bernoulli convolutions and the Φ-variation of functions in the Takagi class
• A probabilistic approach to the Φ-variation of classical fractal functions with critical roughness
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