### Quickest Detection Problems for Bessel Processes

Authors: Peter Johnson, Goran Peskir

Journal: Annals of Applied Probability

Publication Date: 07 February, 2017

Department of: Mathematics

### In Abstract

**Adventures in the mathematics of Brownian motion**

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 when the particle departs from the plane as accurately as possible.

Motivated by industrial applications, researchers from the University of Manchester have solved this problem in the most uncertain scenario when the random/unobservable time of departure 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.

This problem formulation of quickest detection has been extensively studied since the 1960s. In all these problems, however, the signal-to-noise ratio (defined as the difference between the new drift and the old drift divided by the diffusion coefficient) is constant. This is no longer the case in the quickest detection problem studied in this work, and this is thought to be the first time that this problem has been solved.

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