Finite Horizon Decision Timing with Partially Observable Poisson Processes

We study decision timing problems on finite horizon with Poissonian information ar-rivals. 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 in-formation 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 en-vironment. Our method lends itself to simple numerical implementation and we present several illustrative numerical examples. 1.