## The Annals of Applied Probability

### Quickest detection problems for Bessel processes

#### Abstract

Consider the motion of a Brownian particle that initially takes place in a two-dimensional plane and then after some random/unobservable time continues in the three-dimensional space. Given that only the distance of the particle to the origin is being observed, the problem is to detect the time at which the particle departs from the plane as accurately as possible. We solve this problem in the most uncertain scenario when the random/unobservable time is (i) exponentially distributed and (ii) independent from the initial motion of the particle in the plane. The solution is expressed in terms of a stopping time that minimises the probability of a false early detection and the expected delay of a missed late detection.

#### Article information

Source
Ann. Appl. Probab., Volume 27, Number 2 (2017), 1003-1056.

Dates
Revised: March 2016
First available in Project Euclid: 26 May 2017

https://projecteuclid.org/euclid.aoap/1495764373

Digital Object Identifier
doi:10.1214/16-AAP1223

Mathematical Reviews number (MathSciNet)
MR3655860

Zentralblatt MATH identifier
1370.60135

#### Citation

Johnson, Peter; Peskir, Goran. Quickest detection problems for Bessel processes. Ann. Appl. Probab. 27 (2017), no. 2, 1003--1056. doi:10.1214/16-AAP1223. https://projecteuclid.org/euclid.aoap/1495764373

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