Banach spaces of continuous paths with finite $p$-th variation

TL;DR

This paper develops a coefficient-based method using Faber-Schauder expansions to construct and analyze continuous paths with prescribed finite p-th variation, preserving linear and regularity properties.

Contribution

It introduces a novel approach to construct and control paths with specific p-th variation using Faber-Schauder coefficients, extending to broader partition sequences.

Findings

Constructed paths with linear p-th variation from Faber-Schauder coefficients.

Prescribed nonlinear p-th variation via multiplicative transformation.

Showed density of paths with given p-th variation in H"older continuous functions.

Abstract

We study pathwise $p$-th variation of continuous paths on a compact interval along a fixed partition sequence. Although the class of continuous paths with finite $p$-th variation is generally not linear, we develop a coefficient-based approach via Faber-Schauder expansions that, for any $p>1$, enables the construction of paths with prescribed $p$-th variation while preserving useful linear structures and H\"older regularity. We first construct continuous paths with linear $p$-th variation from suitable conditions on their Faber-Schauder coefficients. We then prescribe nonlinear $p$-th variation through a multiplicative transformation and show that, whenever nonempty, the class of H\"older continuous paths with a given $p$-th variation is dense in $C([0,1])$. Next, we introduce a transport procedure that turns a Banach subspace of continuous functions into a Banach subspace of paths with explicitly controlled $p$-th variation. We also prove stability of the associated pathwise F\"ollmer-It\^o map on these transported subspaces. Finally, via time-changes, we show that this constructive framework extends from $q$-adic partition sequences to broader classes of dense $q$-refining partition sequences.