Quadratic variation and quadratic roughness

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.