Contributions to the theory of optimal stopping

Doktorsavhandling, 2010

This thesis deals with the explicit solution of optimal stopping problems with infinite time horizon. To solve Markovian problems in continuous time we introduce an approach that gives rise to explicit results in various situations. The main idea is to characterize the optimal stopping set as the union of the maximum points of explicitly given functions involving the harmonic functions for the underlying stochastic process. This provides elementary solutions for a variety of optimal stopping problems and answers questions concerning the geometric shape of the optimal stopping set. The approach is shown to work well for one- and multidimensional diffusion processes, spectrally negative Lévy processes and problems containing the running maxima process. Furthermore we introduce a new class of problems, which we call problems with guarantee. For continuous one-dimensional driving processes and certain Lévy processes we prove that the optimal strategies are of two-sided type and establish first-order ODEs that characterize the solution. In the second part we consider optimal stopping problems for autoregressive processes in discrete time. This class of processes is intensively studied in statistics and other fields of applied probability. We establish elementary conditions to ensure that the optimal stopping time is of threshold type and find the joint distribution of the threshold-time and the overshoot for a wide class of innovations. Using the principle of continuous fit this leads to explicit solutions.