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 to define invariance notion of stochastic integrals. We introduce the concept of quadratic roughness of a path along a partition sequence and show that for Hölder-continuous paths satisfying this roughness condition, the quadratic variation along balanced partitions is invariant with respect to the choice of the partition sequence. Using these results we derive a formulation of the pathwise Föllmer-Itô calculus which is invariant with respect to the partition sequence.

We further 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