# Quadratic variation and quadratic roughness

## TL;DR

This paper investigates the quadratic variation of continuous paths, introduces quadratic roughness, and demonstrates invariance of quadratic variation and pathwise integration for certain paths, including Brownian motion, across different partitions.

## Contribution

It defines quadratic roughness and proves invariance of quadratic variation and pathwise integration for H"older paths satisfying this condition.

## Key findings

- Quadratic variation along balanced partitions is invariant for paths with quadratic roughness.
- Brownian motion paths satisfy quadratic roughness almost-surely.
- Pathwise integration formulation is invariant under different partition sequences.

## Abstract

We study the concept of quadratic variation of a continuous path along a sequence of partitions and its dependence with respect to the choice of the partition sequence. We define the concept of quadratic roughness of a path along a partition sequence and show that, for H\"older-continuous paths satisfying this roughness condition, the quadratic variation along balanced partitions is invariant with respect to the choice of the partition sequence. Typical paths of Brownian motion are shown to satisfy this quadratic roughness property almost-surely along any partition with a required step size condition. Using these results we derive a formulation of F\"ollmer's pathwise integration along paths with finite quadratic variation which is invariant with respect to the partition sequence.