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We develop an approach for solving one-sided optimal stopping problems indiscrete time for general underlying Markov processes on the real line. Themain idea is to transform the problem into an auxiliary problem for the ladderheight variables. In case that the original problem has a one-sided solutionand the auxiliary problem has a monotone structure, the corresponding myopicstopping time is optimal for the original problem as well. This elementary lineof argument directly leads to a characterization of the optimal boundary in theoriginal problem: The optimal threshold is given by the threshold of the myopicstopping time in the auxiliary problem. Supplying also a sufficient conditionfor our approach to work, we obtain solutions for many prominent examples inthe literature, among others the problems of Novikov-Shiryaev, Shepp-Shiryaev,and the American put in option pricing under general conditions. As a furtherapplication we show that for underlying random walks (and L\'evy processes incontinuous time), the reward functions $g$ leading to one-sided stoppingproblems are exactly the monotone and log-concave functions.

A General Method for Finding the Optimal Threshold in Discrete Time