This paper presents an optimal stopping problem with recall where some cost must be paid to accept the best offer y which has so far appeared. In this model, the optimal decision rule may have a property, called the Double Reservation Value (DRV) property, that for at least one current offer w there exist two different critical values y*, and y* such that, if y*, ⩽y⩽y*, continuing is optimal, or else stopping by accepting the better offer between the current offer w and the best offer y up to the previous point in time is optimal. This property is known to appear in the model with uncertain recall but not in the model with recall. However, we find that it may also appear in the latter model by introducing recall cost. This is one of the most distinctive results in this paper. We examine conditions for the optimal decision rule to have the DRV property in this model and its limiting behavior as the time horizon tends to infinity.
