Upper and Lower Bounds of Optimal Stopping for a Random Sequence: The Case of Finite Horizon

Abstract

This paper derives upper and lower bounds of the price in the optimal stopping problem for a an adapted random sequence in the case of finite horizon. As is demonstrated below, the bounds can be found by solving the maximax and maximin setups of optimal stopping problems. For these setups, we obtain conditions under which 1) a recurrent relation is satisfied for the upper (lower) truncated sequence of optimal stopping prices; 2) an optimality criterion is constructed for the stopping times; 3) the structure and invariance of the optimal stopping times are established. Some examples with explicit solutions of the maximax and maximin setups of optimal stopping problems are given.