Thanks. So what is the condition for a process being arb-free? I.e. so that there is at most one price for our option, rather than (at least) two in the barrier case?

[EDIT] I am guessing that the full answer lies in this literature review by Schachermayer https://www.mat.univie.ac.at/~schacherm ... r0141a.pdf

The issue is always whether or not one can construct a trading strategy with a riskless profit. But, as the article shows, it becomes quite technical in continuous time, continuous state spaces. Personally, I just try to keep in mind "processes with known arbitrage opps". I am usually willing to skip the proofs.