Compound Poisson disorder problem with uniformly distributed disorder time

Abstract

Suppose that the arrival rate and the jump distribution of a compound Poisson process change suddenly at an unknown and unobservable time. We want to detect the change as quickly as possible to take counteractions, e.g., to assure top quality of products in a production system, or to stop credit card fraud in a banking system. If we have no prior information about future disorder time, then we typically assume that the disorder is equally likely to happen any time – or has uniform distribution – over a long but finite time horizon. We solve this so-called compound Poisson disorder problem for the practically important case of unknown, unobserved, but uniformly distributed disorder time. The solution hinges on the complete separation of information flow from the hard time horizon constraint, by describing the former with an autonomous time-homogeneous one-dimensional Markov process in terms of which the detection problem translates into a finite horizon optimal stopping problem. For any given finite horizon, the solution is two-dimensional. For cases where the horizon is large and one is unwilling to set a fixed value for it, we give a one-dimensional approximation. Also, we discuss an extension where the disorder may not happen on the given interval with a positive probability. In this extended model, if no detection decision is made by the end of the horizon, then a second level hypothesis testing problem is solved to determine the local parameters of the observed process.