Quadratic variation along refining partitions: Constructions and Examples

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 $\frac{1}{2}$-Hölder continuous. Finally, we show how to extend these constructions to higher dimensions.

arXiv comment37 pages, 4 figures To appear in: Journal of Mathematical Analysis and Applications