Annals of Applied Probability

Quickest detection problems for Bessel processes

Peter Johnson and Goran Peskir

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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

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

Received: September 2015
Revised: March 2016
First available in Project Euclid: 26 May 2017

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Zentralblatt MATH identifier

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60J60: Diffusion processes [See also 58J65] 60H30: Applications of stochastic analysis (to PDE, etc.)
Secondary: 35K67: Singular parabolic equations 45G10: Other nonlinear integral equations 62C10: Bayesian problems; characterization of Bayes procedures

Quickest detection Brownian motion Bessel process optimal stopping parabolic partial differential equation free-boundary problem smooth fit entrance boundary nonlinear Fredholm integral equation the change-of-variable formula with local time on curves/surfaces


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

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