Sequential Testing Problems for Bessel Processes

Peter Johnson, Goran Peskir

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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.
    Original languageEnglish
    Pages (from-to)2085-2113
    Number of pages29
    JournalTransactions of the American Mathematical Society
    Volume370
    Issue number3
    Early online date8 Sept 2017
    DOIs
    Publication statusPublished - 1 Jan 2018

    Keywords

    • Bessel process
    • Brownian motion
    • Entrance boundary
    • Free-boundary problem
    • Nonlinear Volterra integral equation
    • Optimal stopping
    • Parabolic partial differential equation
    • Sequential testing
    • Signal-to-noise ratio
    • Smooth fit
    • The change-of-variable formula with local time on curves

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