Quadratic variation along refining partitions: Constructions and Examples

TL;DR

This paper develops new methods for constructing paths with finite quadratic variation along refining partitions, analyzing their properties and invariance, including applications to Brownian motion and higher dimensions.

Contribution

It extends previous quadratic variation constructions to non-uniform partitions and identifies classes of paths with invariant quadratic variation under coarsening.

Findings

Paths with quadratic variation invariant under coarsening include Brownian motion paths.

Constructed paths can be $rac{1}{2}$-H"older continuous.

Extensions to higher-dimensional paths are demonstrated.

Abstract

We present several constructions of paths and processes with finite quadratic variation along a refining sequence of partitions, extending previous constructions to the non-uniform case. We study in particular the dependence of quadratic variation with respect to the sequence of partitions for these constructions. We identify a class of paths whose quadratic variation along a partition sequence is invariant under {\it coarsening}. This class is shown to include typical sample paths of Brownian motion, but also paths which are $\frac{1}{2}$-H\"older continuous. Finally, we show how to extend these constructions to higher dimensions.