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Optimal Detection of a Change Point in a Poisson Process for Different Observation Schemes

Abstract.  Change point problems are considered where at some unobservable time the intensity of a point process (Tn), n ∈ 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 (Tn), 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.