C\`adl\`ag Rough Differential Equations with Reflecting Barriers

TL;DR

This paper studies rough differential equations with jumps and reflecting barriers, establishing existence, uniqueness, and stability results for such equations driven by c extasciitilde{}adl extasciitilde{}ag rough paths, expanding the theory of reflected rough differential equations.

Contribution

It introduces a framework for reflected rough differential equations with jumps, proving existence and uniqueness results in new settings involving c extasciitilde{}adl extasciitilde{}ag rough paths.

Findings

Existence of solutions for reflected rough differential equations with jumps.

Uniqueness of solutions in the one-dimensional case.

Stability results for solutions under perturbations.

Abstract

We investigate rough differential equations with a time-dependent reflecting lower barrier, where both the driving (rough) path and the barrier itself may have jumps. Assuming the driving signals allow for Young integration, we provide existence, uniqueness and stability results. When the driving signal is a c\`adl\`ag $p$-rough path for $p \in [2,3)$, we establish existence to general reflected rough differential equations, as well as uniqueness in the one-dimensional case.