Financial/Actuarial Mathematics Seminar

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 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 Föllmer's pathwise integration along paths with finite quadratic variation which is invariant with respect to the partition sequence.

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. Speaker(s): Purba Das (UM)