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Quickest detection problems for Bessel processes. (English) Zbl 1370.60135
Summary: 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.

MSC:
60J60 Diffusion processes
60G40 Stopping times; optimal stopping problems; gambling theory
60H30 Applications of stochastic analysis (to PDEs, etc.)
60J65 Brownian motion
35K67 Singular parabolic equations
35R35 Free boundary problems for PDEs
62C10 Bayesian problems; characterization of Bayes procedures
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