The Annals of Applied Probability

The disorder problem for compound Poisson processes with exponential jumps

Pavel V. Gapeev

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The problem of disorder seeks to determine a stopping time which is as close as possible to the unknown time of “disorder” when the observed process changes its probability characteristics. We give a partial answer to this question for some special cases of Lévy processes and present a complete solution of the Bayesian and variational problem for a compound Poisson process with exponential jumps. The method of proof is based on reducing the Bayesian problem to an integro-differential free-boundary problem where, in some cases, the smooth-fit principle breaks down and is replaced by the principle of continuous fit.

Article information

Ann. Appl. Probab., Volume 15, Number 1A (2005), 487-499.

First available in Project Euclid: 28 January 2005

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Zentralblatt MATH identifier

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 62M20: Prediction [See also 60G25]; filtering [See also 60G35, 93E10, 93E11] 34K10: Boundary value problems
Secondary: 62C10: Bayesian problems; characterization of Bayes procedures 62L15: Optimal stopping [See also 60G40, 91A60] 60J75: Jump processes

Disorder (quickest detection) problem Lévy process, compound Poisson process optimal stopping integro-differential free-boundary problem principles of smooth and continuous fit measure of jumps and its compensator Girsanov’s theorem for semimartingales Itô’s formula


Gapeev, Pavel V. The disorder problem for compound Poisson processes with exponential jumps. Ann. Appl. Probab. 15 (2005), no. 1A, 487--499. doi:10.1214/105051604000000981.

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