Degeneracy condition for the optimal moment in the optimal stopping problem for a new functional of a symmetric random walk and its maximum

Abstract

New classes of functionals are proposed for an optimal stopping problem for a functional of a symmetric random walk and its maximum. For one class the optimal moment in a finite time interval is the beginning of this interval and for another one this is its end. These classes generalize those known previously. A proof of the optimality of the indicated moments is based on combinatorial analysis of random walk trajectories.