Quadratic variation of c\`adl\`ag semimartingales as a.s. limit of the normalized truncated variations

TL;DR

This paper introduces a novel sequence of partition-independent quantities for cadlag paths that almost surely converge to the continuous part of the quadratic variation of semimartingales, offering new insights into pathwise integration.

Contribution

It presents a new approach to define Föllmer's pathwise integral using semi-explicit quantities that converge to quadratic variation without relying on partitions.

Findings

Sequences converge almost surely to the quadratic variation's continuous part.

New method for defining Föllmer's pathwise integral.

Implications for stochastic calculus and pathwise analysis.

Abstract

For a real c\`adl\`ag path $x$ we define sequence of semi-explicit quantities, which do not depend on any partitions and such that whenever $x$ is a path of a c\`adl\`ag semimartingale then these quantities tend a.s. to the continuous part of the quadratic variation of the semimartingale. Next, we derive several consequences of this result and propose a new approach to define F\"ollmer's pathwise integral.