## The Annals of Applied Probability

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

### The disorder problem for compound Poisson processes with exponential jumps

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 28 January 2005

**Permanent link to this document**

http://projecteuclid.org/euclid.aoap/1106922334

**Digital Object Identifier**

doi:10.1214/105051604000000981

**Mathematical Reviews number (MathSciNet)**

MR2115049

**Zentralblatt MATH identifier**

1068.60062

**Subjects**

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

**Keywords**

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

#### Citation

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. http://projecteuclid.org/euclid.aoap/1106922334.