Rough Stochastic Analysis with Jumps

TL;DR

This paper introduces an advanced stochastic sewing lemma for discontinuous control functions, enabling the development of rough stochastic analysis with jumps, including integration, differential equations, and Itô formulas in a cadlag setting.

Contribution

It extends rough stochastic analysis to handle jumps and discontinuities, providing new tools for stochastic calculus with jumps.

Findings

Developed a new stochastic sewing lemma for jumps.

Defined rough stochastic integration in cadlag paths.

Established solutions to rough SDEs with jumps and an Itô formula.

Abstract

We present a new version of the stochastic sewing lemma, capable of handling multiple discontinuous control functions. This is then used to develop a theory of rough stochastic analysis in a c\`adl\`ag setting. In particular, we define rough stochastic integration and establish solutions to rough stochastic differential equations with jump discontinuities in both sources of noise, along with an It\^o formula for stochastic controlled paths with jumps.