The It{\^o}-F\"ollmer formula -- nonstandard cases

TL;DR

This paper extends the Itô-Föllmer formula to cadlag paths with quadratic variation in a weak sense, allowing for more general partitions and less restrictive conditions on jumps and quadratic variation existence.

Contribution

It introduces a generalized version of the Itô-Föllmer formula applicable to broader classes of paths and partitions, relaxing previous assumptions.

Findings

Proves the formula for paths with quadratic variation in a weak sense.

Allows for more general partitions than vanishing mesh partitions.

Accommodates paths with jumps not aligned with quadratic variation jumps.

Abstract

The purpose of this note is to prove the It{\^o}-F\"ollmer formula for the c\`adl\`ag paths possessing quadratic variation in a possibly ``weakest'' sense along some sequence of partitions. By this we mean, for example, that we do not require the jumps of the quadratic variation to occur exactly at the times when the path itself jumps or that the quadratic variation exists at all time instances. Moreover, we deal with more general partitions than partitions with vanishing mesh used by F\"ollmer, also relaxing restrictions imposed on a sequence of partitions in other related literature.