The Annals of Applied Probability

Adaptive Poisson disorder problem

Erhan Bayraktar, Savas Dayanik, and Ioannis Karatzas

Full-text: Open access

Abstract

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.

Article information

Source
Ann. Appl. Probab., Volume 16, Number 3 (2006), 1190-1261.

Dates
First available in Project Euclid: 2 October 2006

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1159804980

Digital Object Identifier
doi:10.1214/105051606000000312

Mathematical Reviews number (MathSciNet)
MR2260062

Zentralblatt MATH identifier
1104.62093

Subjects
Primary: 62L10: Sequential analysis
Secondary: 62L15: Optimal stopping [See also 60G40, 91A60] 62C10: Bayesian problems; characterization of Bayes procedures 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

Keywords
Poisson disorder problem quickest detection optimal stopping

Citation

Bayraktar, Erhan; Dayanik, Savas; Karatzas, Ioannis. Adaptive Poisson disorder problem. Ann. Appl. Probab. 16 (2006), no. 3, 1190--1261. doi:10.1214/105051606000000312. https://projecteuclid.org/euclid.aoap/1159804980


Export citation

References

  • Bayraktar, E. and Dayanik, S. (2006). Poisson disorder problem with exponential penalty for delay. Math. Oper. Res. 31 217–233.
  • Bayraktar, E., Dayanik, S. and Karatzas, I. (2004). Adaptive Poisson disorder problem. Working paper, Princeton Univ. Available at http://www.princeton.edu/~sdayanik/papers/bayes.pdf.
  • Bayraktar, E., Dayanik, S. and Karatzas, I. (2005). The standard Poisson disorder problem revisited. Stochastic Process. Appl. 115 1437–1450.
  • Beibel, M. (1997). Sequential change-point detection in continuous time when the post-change drift is unknown. Bernoulli 3 457–478.
  • Beibel, M. and Lerche, H. R. (2003). Sequential Bayes detection of trend changes. In Foundations of Statistical Inference (Y. Haitovsky, H. R. Lerche and Y. Ritov, eds.) 117–130. Physica, Heidelberg.
  • Brémaud, P. (1981). Point Processes and Queues. Springer, New York.
  • Clarke, F. H., Ledyaev, Y. S., Stern, R. J. and Wolenski, P. R. (1998). Nonsmooth Analysis and Control Theory. Springer, New York.
  • Davis, M. H. A. (1976). A note on the Poisson disorder problem. Banach Center Publ. 1 65–72.
  • Davis, M. H. A. (1993). Markov Models and Optimization. Chapman and Hall, London.
  • Galchuk, L. I. and Rozovskii, B. L. (1971). The disorder problem for a Poisson process. Theory Probab. Appl. 16 729–734.
  • Gugerli, U. S. (1986). Optimal stopping of a piecewise-deterministic Markov process. Stochastics 19 221–236.
  • Krantz, S. G. and Parks, H. R. (2002). The Implicit Function Theorem. Birkhäuser, Boston.
  • Liptser, R. S. and Shiryaev, A. N. (2001). Statistics of Random Processes I. Springer, Berlin.
  • Øksendal, B. (1998). Stochastic Differential Equations. Springer, Berlin.
  • Peskir, G. and Shiryaev, A. N. (2002). Solving the Poisson disorder problem. In Advances in Finance and Stochastics (K. Sandmann and P. J. Schönbucher, eds.) 295–312. Springer, Berlin.
  • Protter, M. H. and Morrey, Jr., C. B. (1991). A First Course in Real Analysis, 2nd ed. Springer, New York.
  • Rockafellar, R. T. (1997). Convex Analysis. Princeton Univ. Press.