VISCOSITY SOLUTIONS OF OPTIMAL STOPPING PROBLEMS FOR FELLER PROCESSES AND THEIR APPLICATIONS

Abstract

This thesis constitutes a research work on deriving viscosity solutions to optimal stopping problems for Feller processes. We present conditions on the process under which the value function is the unique viscosity solution to a Hamilton-Jacobi-Bellman equation associated with a particular operator. More speci cally, assuming that the underlying controlled process is a Feller process, we prove the uniqueness of the viscosity solution. We also apply our results to study several examples of Feller processes. On the other hand, we try to extend our results by iterative optimal stopping methods in the rest of the work. This approach gives a numerical method to approximate the value function and suggest a way of nding the unique viscosity solution associated to the optimal stopping problem. We use it to study several relevant control problems which can reduce to corresponding optimal stopping problems. e.g., an impulse control problem as well as an optimal stopping problem for jump di usions and regime switching processes. In the end, as a complementary, we are trying to construct optimal stopping problems with multiplicative functionals related to a non-conservative Feller semigroup. As a consequence, viscosity solutions were obtained for such kind of constructions.