On the $p^{\text{th}}$ variation of a class of fractal functions

TL;DR

This paper investigates the $p^{ ext{th}}$ variation of certain fractal functions, including classical examples, revealing linear variation behavior and linking it to probabilistic properties and moments of Bernoulli convolutions, with implications for pathwise Itô calculus.

Contribution

It introduces a probabilistic approach to analyze the $p^{ ext{th}}$ variation of fractal functions and derives recursive formulas for moments related to the variation's slope.

Findings

Fractal functions exhibit linear $p^{ ext{th}}$ variation for $p \\ge 1$.

The slope of the variation can be computed from moments of Bernoulli convolutions.

Existence of signed $p^{ ext{th}}$ variation depends on the parity of $p$ and the function construction.

Abstract

The concept of the $p^{\text{th}}$ variation of a continuous function $f$ along a refining sequence of partitions is the key to a pathwise It\^o integration theory with integrator $f$. Here, we analyze the $p^{\text{th}}$ variation of a class of fractal functions, containing both the Takagi--van der Waerden and Weierstra\ss\ functions. We use a probabilistic argument to show that these functions have linear $p^{\text{th}}$ variation for a parameter $p\ge1$, 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 $p^{\text{th}}$ variation can be computed from the $p^{\text{th}}$ 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 $p^{\text{th}}$ variation, which occurs in pathwise It\^o calculus when $p\ge3$ is an odd integer.