Optimal stopping problems and the weak variational inequalites

Abstract

This thesis studies a relationship between solving optimal stopping problems and solving the weak variational inequalites. We prove a theorem that presents conditions under which the weak variatonal inequalities have a unique solution. In a second theorem, we add conditions under which this solution coincides with the value function of a related optimal stopping problem. We also explain a connection between the weak and strong versions of the variational inequalites. Most of the thesis considers problems where the stochastic process is a one dimensional Itô diffusion. A section is added showing that the main results are still valid for more genereal multidimensional jump diffusions. Several examples show how these results can be applied for verifying the solution of stopping problems. The weak variational inequalites allow solving problems where the strong version does not work, since the smoothness conditions on the value function are less strict. Examples from investment theory are used as an application. Here, we develope a general method that is useful for characterizing the solution of certain problems. We briefly discuss the case of combined optimal control and stopping. By adding more conditions, the main results of the thesis can be applied to such more general problems.