# Canonical RDEs and general semimartingales as rough paths

## TL;DR

This paper develops a theory of rough differential equations that includes jumps, introduces a new metric for solution continuity, and shows that general semimartingales can be lifted to rough paths, leading to new limit theorems.

## Contribution

It extends rough path theory to include jump processes and semimartingales, providing new tools and results for stochastic differential equations with discontinuities.

## Key findings

- A new metric ensures solution continuity with respect to rough paths and jumps.
- Semimartingales can be canonically lifted to rough paths.
- Extended BDG inequality leads to novel limit theorems for semimartingale-driven equations.

## Abstract

In the spirit of Marcus canonical stochastic differential equations, we study a similar notion of rough differential equations (RDEs), notably dropping the assumption of continuity prevalent in the rough path literature. A new metric is exhibited in which the solution map is a continuous function of the driving rough path and a so-called path function, which directly models the effect of the jump on the system. In a second part, we show that general multidimensional semimartingales admit canonically defined rough path lifts. An extension of L\'epingle's BDG inequality to this setting is given, and in turn leads to a number of novel limit theorems for semimartingale driven differential equations, both in law and in probability, conveniently phrased via Kurtz-Protter's uniformly-controlled-variations (UCV) condition. A number of examples illustrate the scope of our results.