Finite Horizon Decision Timing with Partially Observable Poisson Processes

Abstract

We study decision timing problems on finite horizon with Poissonian information arrivals. In our model, a decision maker wishes to optimally time her action in order to maximize her expected reward. The reward depends on an unobservable Markovian environment, and information about the environment is collected through a (compound) Poisson observation process. Examples of such systems arise in investment timing, reliability theory, Bayesian regime detection and technology adoption models. We solve the problem by studying an optimal stopping problem for a piecewise-deterministic process, which gives the posterior likelihoods of the unobservable environment. Our method lends itself to simple numerical implementation and we present several illustrative numerical examples.

Keywords:

• Bayesian sequential analysis
• Decision making
• Markov modulated Poisson processes
• Optimal stopping

AMS Subject Classification:

• Primary 62L10
• Secondary 62C10, 62L15, 60G40

ACKNOWLEDGMENTS

The authors would like to thank the editors and anonymous referees for many helpful comments and remarks that improved the presentation in the article.