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.
Publié le : 2005-02-14
Classification:
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,
60G40,
62M20,
34K10,
62C10,
62L15,
60J75
@article{1106922334,
author = {Gapeev, Pavel V.},
title = {The disorder problem for compound Poisson processes with exponential jumps},
journal = {Ann. Appl. Probab.},
volume = {15},
number = {1A},
year = {2005},
pages = { 487-499},
language = {en},
url = {http://dml.mathdoc.fr/item/1106922334}
}
Gapeev, Pavel V. The disorder problem for compound Poisson processes with exponential jumps. Ann. Appl. Probab., Tome 15 (2005) no. 1A, pp. 487-499. http://gdmltest.u-ga.fr/item/1106922334/