I'm trying to implement Andersen and Broadie's dual method for an upper bound (here) of a regular American Put. I understand the process to compute it, but I have a conceptual issue : everything relies on the computation of an approximation of the Doob's martingale.
This approximation can be computed through Monte-carlo sub-simulations using a stopping rule to determine where to start and stop sub paths. So far, so good. But when looking for the stopping rule for the American Put (there, section 2.3.1), I see that optimal exercises times are defined as the instants when (K-S)+ is above the value of the Put... which seems like going around in circles to me (to compute the put you need the exercise times, to get the exercise times you need the put). So how to concretely proceed ?
(Others use regression to compute the conditional expectations, but since Andersen and Broadie don't, I was wondering whether we could avoid it in this case)