With few exceptions, optimal stopping assumes that the underlying system is stopped immediately after the decision is made.
In fact, most stoppings take time. This has been variously referred to as "time-to-build", "investment lag" and "gestation period",
which is often non negligible.
In this talk, we consider a class of optimal stopping/switching problems with delivery lags, or equivalently, delayed information,
by using reflected BSDE method. As an example, we study American put option with delayed exercise, and show that it can be decomposed
as a European put option and a premium, the latter of which involves a new optimal stopping problem where the investor decides when to stop
to collect the Greek theta of such a European option. We also give a complete characterization of the optimal exercise boundary by resorting to free boundary analysis.
Joint work with Zhou Yang and Mihail Zervos.