Rough semimartingales and $p$-variation estimates for martingale transforms

TL;DR

This paper develops new $p$-variation estimates for martingale-related processes, introduces rough semimartingales as a unifying framework, and advances integration theory relevant to rough paths, stochastic, and harmonic analysis.

Contribution

It introduces a new scale of $p$-variation estimates and the concept of rough semimartingales, unifying classical and rough path theories.

Findings

Established new $p$-variation estimates for martingale transforms

Introduced rough semimartingales and their integration theory

Connected classical semimartingales with rough paths

Abstract

We establish a new scale of $p$-variation estimates for martingale paraproducts, martingale transforms, and It\^o integrals, of relevance in rough paths theory, stochastic, and harmonic analysis. As an application, we introduce rough semimartingales, a common generalization of classical semimartingales and (controlled) rough paths, and their integration theory.