Open Access
February 2005 The disorder problem for compound Poisson processes with exponential jumps
Pavel V. Gapeev
Ann. Appl. Probab. 15(1A): 487-499 (February 2005). DOI: 10.1214/105051604000000981


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.


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Pavel V. Gapeev. "The disorder problem for compound Poisson processes with exponential jumps." Ann. Appl. Probab. 15 (1A) 487 - 499, February 2005.


Published: February 2005
First available in Project Euclid: 28 January 2005

zbMATH: 1068.60062
MathSciNet: MR2115049
Digital Object Identifier: 10.1214/105051604000000981

Primary: 34K10 , 60G40 , 62M20
Secondary: 60J75 , 62C10 , 62L15

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

Rights: Copyright © 2005 Institute of Mathematical Statistics

Vol.15 • No. 1A • February 2005
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