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Transactions of the American Mathematical Society

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Sequential testing problems for Bessel processes


Authors: Peter Johnson and Goran Peskir
Journal: Trans. Amer. Math. Soc. 370 (2018), 2085-2113
MSC (2010): Primary 60G40, 60J60, 60H30; Secondary 35K10, 45G10, 62C10
DOI: https://doi.org/10.1090/tran/7068
Published electronically: September 8, 2017
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Abstract: Consider the motion of a Brownian particle that takes place either in a two-dimensional plane or in three-dimensional space. Given that only the distance of the particle to the origin is being observed, the problem is to detect the true dimension as soon as possible and with minimal probabilities of the wrong terminal decisions. We solve this problem in the Bayesian formulation under any prior probability of the true dimension when the passage of time is penalised linearly.


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Additional Information

Peter Johnson
Affiliation: School of Mathematics, The University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom
Email: peter.johnson-3@manchester.ac.uk

Goran Peskir
Affiliation: School of Mathematics, The University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom
Email: goran@maths.man.ac.uk

DOI: https://doi.org/10.1090/tran/7068
Keywords: Sequential testing, Brownian motion, Bessel process, optimal stopping, parabolic partial differential equation, free-boundary problem, smooth fit, entrance boundary, nonlinear Volterra integral equation, the change-of-variable formula with local time on curves, signal-to-noise ratio.
Received by editor(s): June 7, 2016
Received by editor(s) in revised form: September 8, 2016
Published electronically: September 8, 2017
Additional Notes: This research was supported by a grant from the British Engineering and Physical Sciences Research Council (EPSRC)
Article copyright: © Copyright 2017 by the authors

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