An effective method for the explicit solution of sequential problems on the real line

ABSTRACT

A general method for solving optimal stopping problems for continuous-time Markov processes on the real line is developed. The basic idea is to first study an auxiliary problem for the two-dimensional process consisting of the underlying Markov process and its running maximum. It turns out that this auxiliary problem is much easier to solve using standard methods such as the monotonicity of the problem, and an optimal strategy in the class of threshold times can be found. The optimality of the time carries over to original problem. This two-step procedure gives a unifying approach for solving problems from different fields such as mathematical finance and sequential analysis.

KEYWORDS:

• Generalized parking problems
• Lorden’s test
• Markov processes
• monotone stopping rules
• optimal stopping
• running maximum process
• threshold times

MATHEMATICS SUBJECT CLASSIFICATION:

• 60G40
• 62L10
• 60J25

Acknowledgments

I would like to thank the Editor, Professor Nitis Mukhopadhyay, for inviting me to submit this article and for his careful reading, which greatly improved the presentation of the article. Furthermore, I would like to thank Professor Paavo Salminen for useful remarks.