An optimal stopping problem with two levels of incomplete information

The following variant of a classical selection problem is considered. Offers of random size are successively available to a decision-maker at discrete time instants, and he has to select one without having the possibility of recall. At any time he can accept or reject the current offer based on some initial information (just received) on its still unknown size, or he can demand additional information and take his decision thereafter. The costc0 > 0 for continuing to the next offer and the costd0 > 0 for obtaining the additional information are assumed to be constant. Under reasonable conditions on the underlying stochastic structure we find and characterize strategies maximizing the expected net reward.