A superhedging approach to stochastic integration

TL;DR

This paper establishes the existence of quadratic variation for typical price paths using a superhedging approach and develops a model-free Itô integration, advancing stochastic analysis in financial mathematics.

Contribution

It introduces a superhedging-based method to prove quadratic variation existence and constructs a model-free Itô integral for a broad class of paths.

Findings

Quadratic variation exists for typical paths with mild jump restrictions.

The approach applies to non-negative c extquoteright{}adl extquoteright{}ag paths.

A robust, model-free Itô integration framework is developed.

Abstract

Using Vovk's outer measure, which corresponds to a minimal superhedging price, the existence of quadratic variation is shown for "typical price paths" in the space of c\`adl\`ag functions possessing a mild restriction on the jumps directed downwards. In particular, this result includes the existence of quadratic variation of "typical price paths" in the space of non-negative c\`adl\`ag paths and implies the existence of quadratic variation in the sense of F\"ollmer quasi surely under all martingale measures. Based on the robust existence of the quadratic variation, a model-free It\^o integration is developed.