General optimal stopping with linear cost

Abstract

This article treats both discrete time and continuous time stopping problems for general Markov processes on the real line with general linear costs as they naturally arise in many problems in sequential decision making. Using an auxiliary function of maximum representation type, conditions are given to guarantee the optimal stopping time to be of threshold type. The optimal threshold is then characterized as the root of that function. For random walks, our results condense in the fact that all combinations of concave increasing payoff functions and convex cost functions lead to a one-sided solution. For Lévy processes, an explicit way to obtain the auxiliary function and the threshold is given by use of the ladder height processes. Lastly, the connection from discrete and continuous problems and possible approximation of the latter via the former is discussed and the results are applied to sequential tests of power one.