Dubins, L. EShepp, Larry AShiryaev, A. N2023-05-232023-05-2319942016-07-27https://repository.upenn.edu/handle/20.500.14332/47786We 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 → ∞.Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.Bessel processesoptimal stopping rulesmaximal inequalitiesmoving boundary problem for parabolic equations (Stephan problem)local martingalessemimartingalesDirichlet processeslocal timeprocesses with reflectionBrownian motion with drift and reflectionProbabilityStatistics and ProbabilityArticle