On an effective solution of the optimal stopping problem for random walks

We find a solution of the optimal stopping problem for the case when a reward function is an integer power function of a random walk on an infinite time interval. It is shown that an optimal stopping time is a first crossing time through a level defined as the largest root of Appell's polynomial associated with the maximum of the random walk. It is also shown that a value function of the optimal stopping problem on the finite interval $\{0,1\ldots T\}$ converges with an exponential rate as $T\to\infty$ to the limit under the assumption that jumps of the random walk are exponentially bounded.