Quickest search over Brownian channels

Abstract

In this paper we resolve an open problem proposed by Lai, Vincent Poor, Xin, and Georgiadis [Quickest search over multiple sequences. IEEE Trans. Inf. Theory 57(8) (2011), pp. 5375–5386]. Consider a sequence of Brownian motions with unknown drift equal to one or zero, which may be observed one at a time. We give a procedure for finding, as quickly as possible, a process which is a Brownian motion with non-zero drift. This original quickest search problem, in which the filtration itself is dependent on the observation strategy, is reduced to a single filtration impulse control and optimal stopping problem, which is in turn reduced to an optimal stopping problem for a reflected diffusion, which can be explicitly solved.

Keywords::

• Bayesian quickest search
• optimal switching
• optimal stopping
• reflected diffusion

Acknowledgements

This work was supported by the National Science Foundation under Grant DMS-1118673. The authors would like to thank the referees, associate editor and editor for their helpful comments, which helped us greatly improve the paper.

Notes

^1. Email: [email protected]