On pathwise quadratic variation for cadlag functions

TL;DR

This paper refines the concept of pathwise quadratic variation for cadlag functions by linking it to Skorokhod topology, leading to a more robust and simplified definition that naturally includes Lebesgue decomposition.

Contribution

It introduces a reformulation of quadratic variation as a Skorokhod topology limit, simplifying the concept and ensuring it encompasses Lebesgue decomposition inherently.

Findings

Quadratic variation can be defined as a Skorokhod topology limit.

The new definition simplifies previous approaches.

Lebesgue decomposition follows naturally from the new formulation.

Abstract

We revisit H. Foellmer's concept of quadratic variation of a cadlag function along a sequence of time partitions and discuss its relation with the Skorokhod topology. We show that in order to obtain a robust notion of pathwise quadratic variation applicable to sample paths of cadlag processes, one must reformulate the definition of pathwise quadratic variation as a limit in Skorokhod topology of discrete approximations along the partition. One then obtains a simpler definition of quadratic variation which implies the Lebesgue decomposition as a result, rather than requiring it as an extra condition.