• Bernoulli
  • Volume 3, Number 4 (1997), 457-478.

Sequential change-point detection in continuous time when the post-change drift is unknown

M. Beibel

Full-text: Open access


Let Wt(0≤t<∞) denote a Brownian motion process which has zero drift during the time interval [0,ν) and drift θ during the time interval [ν,∞), where θ and ν are unknown. The process W is observed sequentially. The general goal is to find a stopping time T of W that 'detects' the unknown time point ν as soon and as reliably as possible on the basis of this information. Here stopping always means deciding that a change in the drift has already occurred. We discuss two particular loss structures in a Bayesian framework. Our first Bayes risk is closely connected to that of the Bayes tests of power one of Lerche. The second Bayes risk generalizes the disruption problem of Shiryayev to the case of unknown θ.

Article information

Bernoulli, Volume 3, Number 4 (1997), 457-478.

First available in Project Euclid: 6 April 2007

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Bayes problems Brownian motion change point sequential detection tests of power one


Beibel, M. Sequential change-point detection in continuous time when the post-change drift is unknown. Bernoulli 3 (1997), no. 4, 457--478.

Export citation


  • [1] Beibel, M. (1994a) Bayes-Optimalität in Trendänderungsmodellen mit kontinuierlicher Zeit. Ph.D. thesis, Albert-Ludwigs Universität, Freiburg.
  • [2] Beibel, M. (1994b) Bayes problems in change-point models for the Wiener process. In E. Carlstein, H.G. Mü ller and D. Siegmund (eds), Change-point Problems, pp. 1-6. Lecture Notes Monograph Series 23. Hayward, CA: Institute of Mathematical Statistics.
  • [3] Beibel, M. (1996) A note on Ritov's Bayes approach to the minimax property of the cusum procedure. Ann. Statist., 24, 1804-1812.
  • [4] Beibel, M. and Lerche, H.R. (1997) A new look at optimal stopping problems related to mathematical finance. Statist. Sinica., 7, 93-108.
  • [5] Darling, D. and Robbins, H.R. (1967) Iterated logarithm inequalities. Proc. Nat. Acad. Sci. USA, 57, 1188-1192.
  • [6] Keener, R., Lerche, H.R. and Woodroofe, M. (1995) A nonlinear parking problem. Sequential Anal., 14, 247-272.
  • [7] Kohler, O. (1995) Erwartete Verzögerungen bei Trenderkennung in Bayes-Modellen mit Brownscher Bewegung. Master's thesis, Albert-Ludwigs Universität, Freiburg.
  • [8] Lerche, H.R. (1986a) An optimality property of the repeated significance test. Proc. Nat. Acad. Sci. USA, 83, 1546-1548.
  • [9] Lerche, H.R. (1986) The shape of Bayes tests of power one. Ann. Statist., 14, 1030-1048.
  • [10] Liptser, R.S. and Shiryayev, A.N. (1977) Statistics of Random Processes, Vol. 1. Berlin: Springer- Verlag.
  • [11] Pollak, M. (1987) Average run lengths of an optimal method of detecting a change in distribution. Ann. Statist., 15, 749-779.
  • [12] Pollak, M. and Siegmund, D. (1975) Approximations to the expected sample size of certain sequential tests. Ann. Statist., 3, 1267-1282.
  • [13] Pollak, M. and Siegmund, D. (1985) A diffusion process and its application to detecting a change in the drift of Brownian motion. Biometrika, 72, 267-280.
  • [14] Ritov, Y. (1990) Decision theoretic optimality of the cusum procedure. Ann. Statist., 18, 464-1469.
  • [15] Shiryayev, A.N. (1963) On optimum methods in quickest detection problems. Theory Probab. Appl., 8, 22-46.
  • [16] Shiryayev, A.N. (1973) Statistical Sequential Analysis. Translations of Mathematical Monographs 8. Providence, RI: American Mathematical Society.
  • [17] Woodroofe, M., Lerche, R. and Keener, R. (1994) A generalized parking problem. In S.S. Gupta and J.O. Berger (eds), Statistical Decision Theory and Related Topics V, pp. 523-532. Berlin: Springer-Verlag.
  • [18] Yor, M. (1992) Sur certaines fonctionelles exponentielles du mouvement brownien réel. J. Appl. Probab., 29, 202-208.