On (signed) Takagi-Landsberg functions: $p^{\text{th}}$ variation, maximum, and modulus of continuity

TL;DR

This paper investigates signed Takagi-Landsberg functions with Hurst parameter H, establishing their p-th variation, analyzing their maximum and modulus of continuity, and exploring their applications in pathwise Itô calculus.

Contribution

It introduces the class rak X^H of functions, proves their linear p-th variation, and characterizes their extremal properties and regularity measures.

Findings

Functions in rak X^H have linear p-th variation with p=1/H.

The slope of variation relates to the p-th absolute moment of Bernoulli convolutions.

The paper determines the maximum, oscillation, and modulus of continuity for these functions.

Abstract

We study a class $\mathfrak X^H$ of signed Takagi-Landsberg functions with Hurst parameter $H\in(0,1)$. We first show that the functions in $\mathfrak X^H$ admit a linear $p^{\text{th}}$ variation along the sequence of dyadic partitions of $[0,1]$, where $p=1/H$. The slope of the linear increase can be represented as the $p^{\text{th}}$ absolute moment of the infinite Bernoulli convolution with parameter $2^{H-1}$. The existence of a continuous $p^{\text{th}}$ variation enables the use of the functions in $\mathfrak X^H$ as test integrators for higher-order pathwise It\^o calculus. Our next results concern the maximum, the maximizers, and the modulus of continuity of the classical Takagi-Landsberg function for all $0<H<1$. Then we identify the uniform maximum, the uniform maximal oscillation, and a uniform modulus of continuity for the class $\mathfrak X^H$.