  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.
