On the sequential testing problem for some diffusion processes

Abstract

We study the Bayesian problem of sequential testing of two simple hypotheses about the drift rate of an observable diffusion process. The optimal stopping time is found as the first time at which the posterior probability of one of the hypotheses exits a region restricted by two stochastic boundaries depending on the current observations. The proof is based on an embedding of the initial problem into a two-dimensional optimal stopping problem and the analysis of the associated parabolic-type free-boundary problem. We also show that the problem admits a closed-form solution under certain non-trivial relations between the coefficients of the observable diffusion.

Keywords:

• sequential testing
• diffusion process
• two-dimensional optimal stopping
• stochastic boundary
• parabolic-type free-boundary problem
• a change-of-variable formula with local time on surfaces

2000 Mathematics Subject Classification::

• primary 60G40
• 62M20
• 34K10
• secondary 62C10
• 62L15
• 60J60

Acknowledgements

The authors thank the Referee for his careful reading of the manuscript and useful comments. This research was partially supported by Deutsche Forschungsgemeinschaft through the SFB 649 Economic Risk.

Notes

1. [email protected]

Additional information

Notes on contributors

Albert N. Shiryaev

1 1. [email protected]