On the quadratic variation of the model-free price paths with jumps

TL;DR

This paper demonstrates that certain model-free cadlag price paths with controlled jumps have a well-defined quadratic variation independent of the partition sequence, and introduces partition-independent quantities converging to it.

Contribution

It establishes the existence of partition-independent quadratic variation for model-free price paths with jumps and defines related convergent quantities.

Findings

Quadratic variation exists independently of partition sequences.

Price paths with mild downward jumps have well-defined quadratic variation.

Partition-independent quantities can be used to approximate quadratic variation.

Abstract

We prove that the model-free typical (in the sense of Vovk) c\`adl\`ag price paths with mildly restricted downward jumps possess quadratic variation which does not depend on the specific sequence of partitions as long as these partitions are obtained from stopping times such that the oscillations of a path on the consecutive (half-open on the right) intervals of these partitions tend (in a specified sense) to 0. Finally, we also define quasi-explicit, partition independent quantities which tend to this quadratic variation.