What does it mean to say that we "replicate an American option" in the usual structure of an asset-pricing model with a measure Q?
I mean, any portfolio with an adapted and self-financing strategy will be a martingale. But the payoff of an American option will in general not be a martingale.
So when people talk about replicating an American option, what do they really mean? Do they mean replicating an American option's payoff assuming optimal behavior? So really we try to replicate the stopped payoff process?