Sequential Testing Problems for Lévy Processes

Abstract

We present the sequential testing of two simple hypotheses for a large class of Lévy processes. As usual in this framework, the initial optimal stopping problem is reduced to a free-boundary problem, solved through the principles of the smooth and/or continuous fit. The well-known solutions of the Wiener and the Poisson sequential testing can be derived from our procedure. The exact solution for sequentially testing two simple hypotheses concerning the parameter p, 0 < p < 1, of a negative binomial process is explicitly given.

Keywords:

• Bayes decision rule
• Free-boundary Stephan problem
• Lévy processes
• Negative binomial process
• Optimal stopping
• Principles of continuous and smooth fit
• Sequential testing

Subject Classifications:

• 60G40
• 62C10
• 62L10
• 34K10

ACKNOWLEDGMENTS

The authors are grateful to G. Peskir for his interest and precious remarks and to N. Mukhopadhyay for a useful discussion, held when he visited Bocconi University in March 2012. The authors also thank the Associate Editor and the referee, who have examined the article.

Notes

Recommended by N. Mukhopadhyay