On a class of generalized Takagi functions with linear pathwise quadratic variation

TL;DR

This paper studies a class of continuous functions related to generalized Takagi functions, highlighting their quadratic variation properties and implications for pathwise calculus, including uniform bounds and examples of non-additive quadratic variation.

Contribution

It characterizes the quadratic variation and uniform properties of a new class of functions related to Takagi functions, with implications for pathwise stochastic calculus.

Findings

Computed the pointwise maximum and oscillation of functions in the class

Established the uniform modulus of continuity for all functions in the class

Provided an example where the quadratic variation of a sum does not exist

Abstract

We consider a class $\mathscr{X}$ of continuous functions on $[0,1]$ that is of interest from two different perspectives. First, it is closely related to sets of functions that have been studied as generalizations of the Takagi function. Second, each function in $\mathscr{X}$ admits a linear pathwise quadratic variation and can thus serve as an integrator in F\"ollmer's pathwise It\=o calculus. We derive several uniform properties of the class $\mathscr{X}$. For instance, we compute the overall pointwise maximum, the uniform maximal oscillation, and the exact uniform modulus of continuity for all functions in $\mathscr{X}$. Furthermore, we give an example of a pair $x,y\in\mathscr{X}$ such that the quadratic variation of the sum $x+y$ does not exist.