Poisson disorder problem with control on costly observations

A Poisson process Xt changes its rate at an unknown and unobservable time θ from λ0 to λ1. Detecting the change time as quickly as possible in an optimal way is described in literature as the Poisson disorder problem. We provide a more realistic generalization of the disorder problem for Poisson process by introducing fixed and continuous costs for being able to observe the arrival process. As a result, in addition to finding the optimal alarm time, we also characterize an optimal way of observing the arrival process. We illustrate the structure of the solution spaces with the help of some numerical examples.