Differential equations driven by rough paths with jumps

TL;DR

This paper extends rough path theory to include stochastic differential equations driven by semimartingales with jumps, broadening the scope of pathwise solutions for stochastic calculus.

Contribution

It introduces a rough path framework for Itô stochastic integration and differential equations involving semimartingales with jumps, expanding the applicability of pathwise methods.

Findings

Developed a rough path approach for jump-driven stochastic equations

Enhanced the class of stochastic processes treatable with pathwise techniques

Discussed multiple applications of the extended theory

Abstract

We develop the rough path counterpart of It\^o stochastic integration and - differential equations driven by general semimartingales. This significantly enlarges the classes of (It\^o / forward) stochastic differential equations treatable with pathwise methods. A number of applications are discussed.