We study a general optimal stopping problem for a strong Markov process in the case when there is a time lag $\delta>0$ from the time $\tau$ when the decision to stop is taken (a stopping time) to the time $\tau+\delta$ when the system actually stops. Equivalently, we impose the constraint that the admissible times for stopping are stopping (Markov) times with respect to the delayed flow of information. It is shown that such a problem can be reduced to a classical optimal stopping problem by a simple transformation. The results are applied
(i) to find the optimal time to sell an asset
(ii) to solve an optimal resource extraction problem,
in both cases under delayed information flow.