  We study a continuous time Bayesian quickest detection problem in which
observation times are a scarce resource. The agent, limited to making a finite
number of discrete observations, must adaptively decide his observation
strategy to minimize detection delay and the probability of false alarm. Under
two different models of observation rights, we establish the existence of
optimal strategies, and formulate an algorithmic approach to the problem via
jump operators. We describe algorithms for these problems, and illustrate them
with some numerical results. As the number of observation rights tends to
infinity, we also show convergence to the classical continuous observation
problem of Shiryaev.
