Non-uniqueness for reflected rough differential equations

Abstract

We give an example of a reflected diffferential equation which may have infinitely many solutions if the driving signal is rough enough (e.g. of infinite $p$-variation, for some $p>2$). For this equation, we identify a sharp condition on the modulus of continuity of the signal under which uniqueness holds. L\'evy's modulus for Brownian motion turns out to be a boundary case. We further show that in our example, non-uniqueness holds almost surely when the driving signal is a fractional Brownian motion with Hurst index $H < \frac{1}{2}$. The considered equation is driven by a two-dimensional signal with one component of bounded variation, so that rough path theory is not needed to make sense of the equation.