Upper and lower bounds of optimal stopping for a random sequence: the case of finite horizon

There in the article upper and lower bounds of value function in optimal stopping problem for adapted random sequence in final time horizon case is established. It has been proved, that to find these bounds one have to solve maximax and maximin optimal stopping problems. For the problems conditions have been found, under which: 1) upper (lower) truncated sequence of values satisfies recurrent relation; 2) criterion of optimality for stopping rules holds true; 3) structure and invariance of optimal stopping moments have been established. There are also in the article examples of explicit solutions for above stated extremal optimal stopping problems.