On the martingale and free-boundary approaches in sequential detection problems with exponential penalty for delay

Abstract

We study the connection between the martingale and free-boundary approaches in sequential detection problems for the drift of a Brownian motion, under the assumption of exponential penalty for the delay. By means of the solution of a suitable free-boundary problem, we show that the reward process can be decomposed into the product between a gain function of the boundary point and a positive martingale inside the continuation region.

Keywords::

• optimal stopping
• sequential detection and disorder problems
• free-boundary problem
• martingale approach of Beibel

AMS Subject Classification::

• 60G40
• 60G44
• 35R35

Acknowledgements

We thank the Editor, the Associate Editor and the referees for their precious comments.