Optimal stopping problems with expectation constraints

Abstract

In this thesis we investigate optimal stopping problems with expectation cost constraints. We focus on reducing the set of stopping times as well as on deriving a partial differential equation for the value function. If the process to stop is a time-homogeneous Ito-process, we show, by introducing a new state variable, that one can transform the problem into an unconstrained control problem and hence obtain a dynamic programming principle. We characterize the value function in terms of the dynamic programming equation, which turns out to be an elliptic, fully non-linear partial differential equation of second order. In addition, we prove a classical verification theorem and apply it to several examples. Furthermore, we consider the problem of optimally stopping a one-dimensional regular continuous strong Markov process with a stopping time satisfying an expectation constraint. We show that it is sufficient to consider only stopping times such that the law of the process at the stopping time is a weighted sum of 3 Dirac measures. The proof uses results on Skorokhod embeddings in order to reduce the stopping problem to a linear optimization problem over a convex set of probability measures. We apply the results to analyze a sequential testing problem and show that in this problem the optimal stopping times are given by at most two consecutive exit times of intervals. Finally, using the theory of Tchebycheff systems we examine when we can reduce the set of stopping times in the constrained problem to first exit times of intervals. In this case, the law of the process at the stopping time is a weighted sum of at most 2 Dirac measures.