A harmonic function technique for the optimal stopping of diffusions

Abstract

We consider problems of optimal stopping where the driving process is a (one- or multi-dimensional) diffusion. Our approach is motivated by a change of measure techniques and gives a characterization of the optimal stopping set in terms of harmonic functions for one-dimensional diffusions. The generalization to multidimensional diffusions uses the theory of Martin boundaries. Various applications, including exchange options, are given. We treat an example where halfspaces, which are plausible candidates for the optimal stopping set, are in fact strict subsets of it.

Keywords:

• optimal stopping
• diffusions
• harmonic functions
• Martin boundary
• exchange options

Acknowledgement

We thank the Associated Editor and the referee for their thoughtful and valuable comments.