Contributions to Optimal Stopping and Long-Term Average Impulse Control
In this thesis we consider undiscounted, infinite time horizon optimal stopping problems with generalized linear costs and long-term average impulse control problems. The main goal is to find (semi-)explicit solutions in case the underlying process contains jumps. In order to solve the stopping problems, we utilize embedded monotone problems to find sufficient conditions, that are easy to handle, for a threshold time to be optimal. Further, we characterize the threshold for one-dimensional Markov processes in both discrete and continuous time. While in the discrete time case the concept of ladder times can be used to exploit inherent monotone structures, in continuous time we develop an integral type maximum representation to enable a comparable line of argument. The findings on long-term average impulse control problems are structured in two main areas. First, for a general one-dimensional Markov process we characterize the problem’s value and possible optimal strategies by an associated stopping problem. Then, we develop a step-by-step solution technique in case the process is a Lévy process and demonstrate its usefulness by applying it to relevant examples, among others problems from inventory control and optimal harvesting. Apart from these direct applications we use our theoretical findings to investigate the influence of varying fixed costs on the impulse control problem, study a control problem with a restriction to the impulse frequency and treat mean field games and problems of impulse control.