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Theory of Probability & Its Applications
We consider, for Bessel processes X ∈ Besα with arbitrary order (dimension) α ∈ R, the problem of the optimal stopping (1.4) for which the gain is determined by the value of the maximum of the process X and the cost which is proportional to the duration of the observation time. We give a description of the optimal stopping rule structure (Theorem 1) and the price (Theorem 2). These results are used for the proof of maximal inequalities of the type
E max Xrr≤r ≤ γ(α) is a constant depending on the dimension (order) α. It is shown that γ(α) ∼ √α at α → ∞.
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Bessel processes, optimal stopping rules, maximal inequalities, moving boundary problem for parabolic equations (Stephan problem), local martingales, semimartingales, Dirichlet processes, local time, processes with reflection, Brownian motion with drift and reflection
Dubins, L. E., Shepp, L. A., & Shiryaev, A. N. (1994). Optimal Stopping Rules and Maximal Inequalities for Bessel Processes. Theory of Probability & Its Applications, 38 (2), 226-261. http://dx.doi.org/10.1137/1138024
Date Posted: 27 November 2017
This document has been peer reviewed.