Continuation and uniqueness beyond the corner time

Determine whether the deterministic perturbed Skorokhod equation with a time-dependent boundary can be continued beyond the first time at which its running maximum meets the boundary, and, if a continuation exists, establish whether it is unique and whether its lifetime is finite.

Background

For a positive initial state, the paper constructs the deterministic perturbed Skorokhod solution uniquely only up to the time $\sigma_*:=\inf\{t>0:M_t(w)=b(t)\}$, when the running maximum reaches the time-dependent reflecting boundary. The authors explicitly identify the unresolved issues of extending the solution beyond this corner time, uniqueness of any such extension, and finiteness of its lifetime. The subsequent lemma shows that any solution with finite lifetime can be continuously extended to that lifetime, while later results establish global continuation under the paper’s regularity condition (PB); the quoted passage records the unresolved question before those results are proved.