Open Access
August 2006 Adaptive Poisson disorder problem
Erhan Bayraktar, Savas Dayanik, Ioannis Karatzas
Ann. Appl. Probab. 16(3): 1190-1261 (August 2006). DOI: 10.1214/105051606000000312


We study the quickest detection problem of a sudden change in the arrival rate of a Poisson process from a known value to an unknown and unobservable value at an unknown and unobservable disorder time. Our objective is to design an alarm time which is adapted to the history of the arrival process and detects the disorder time as soon as possible.

In previous solvable versions of the Poisson disorder problem, the arrival rate after the disorder has been assumed a known constant. In reality, however, we may at most have some prior information about the likely values of the new arrival rate before the disorder actually happens, and insufficient estimates of the new rate after the disorder happens. Consequently, we assume in this paper that the new arrival rate after the disorder is a random variable.

The detection problem is shown to admit a finite-dimensional Markovian sufficient statistic, if the new rate has a discrete distribution with finitely many atoms. Furthermore, the detection problem is cast as a discounted optimal stopping problem with running cost for a finite-dimensional piecewise-deterministic Markov process.

This optimal stopping problem is studied in detail in the special case where the new arrival rate has Bernoulli distribution. This is a nontrivial optimal stopping problem for a two-dimensional piecewise-deterministic Markov process driven by the same point process. Using a suitable single-jump operator, we solve it fully, describe the analytic properties of the value function and the stopping region, and present methods for their numerical calculation. We provide a concrete example where the value function does not satisfy the smooth-fit principle on a proper subset of the connected, continuously differentiable optimal stopping boundary, whereas it does on the complement of this set.


Download Citation

Erhan Bayraktar. Savas Dayanik. Ioannis Karatzas. "Adaptive Poisson disorder problem." Ann. Appl. Probab. 16 (3) 1190 - 1261, August 2006.


Published: August 2006
First available in Project Euclid: 2 October 2006

zbMATH: 1104.62093
MathSciNet: MR2260062
Digital Object Identifier: 10.1214/105051606000000312

Primary: 62L10
Secondary: 60G40 , 62C10 , 62L15

Keywords: Optimal stopping , Poisson disorder problem , quickest detection

Rights: Copyright © 2006 Institute of Mathematical Statistics

Vol.16 • No. 3 • August 2006
Back to Top