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

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.