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 coarsening. This class is shown to include typical sample paths of Brownian motion, but also paths which are 1 2 -Hölder continuous. Finally, we show how to extend these constructions to higher dimensions.

The Hurst roughness exponent and its model-free estimation

“…In this case, the standard Faber-Schauder expansion can no longer be applied. It might be interesting to see, though, if the techniques developed by Cont and Das [1,2] can be set to work here.…”

“…Finally, the appendices contain extensive summary statistics of our various estimators. 2 The Hurst roughness exponent: definition and a consistency result…”