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

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.