Optimal detection of a change point in a poisson process for different observation schemes

Change point problems are considered where at some unobservable time the intensity of a point process (T-n), n is an element of N, has a jump. For a given reward functional we detect the change point optimally for different information schemes. These schemes differ in the available information. We consider three information levels, namely sequential observation of (T-n), ex post decision after observing the point process up to a fixed time t* and a combination of both observation schemes. In all of these cases the detection problem is viewed as an optimal stopping problem which can be solved by deriving a semimartingale representation of the gain process and applying tools from filtering theory.