Dubins, L. E.; Shepp, L. A.; Shiryaev, A. N. Optimal stopping rules and maximal inequalities for Bessel processes. (English. Russian original) Zbl 0807.60040 Theory Probab. Appl. 38, No. 2, 226-261 (1993); translation from Teor. Veroyatn. Primen. 38, No. 2, 288-330 (1993). Summary: We consider, for Bessel processes \(X \in \text{Bes}^ \alpha(x)\) with arbitrary order (dimension) \(\alpha \in \mathbb{R}\), the problem of the optimal stopping \[ V^ \alpha_ * (x,s) = \sup_{\tau} {\mathbf E} [S_ \tau (x,s) - c\tau],\quad c\text{ positive constant,} \] for which the gain is determined by the value of the maximum of the process \(X\) and the cost is proportional to the duration of the observation time. We give a description of the optimal stopping rule structure and the price. These results are used for the proof of maximal inequalities of the type \[ {\mathbf E} \max_{r \leq \tau} X_ r \leq \gamma (\alpha) \sqrt{{\mathbf E} \tau}, \] where \(X \in \text{Bes}^ \alpha (0)\), \(\tau\) is arbitrary stopping time, \(\gamma(\alpha)\) is a constant depending on the dimension (order) \(\alpha\). It is shown that \(\gamma(\alpha) \sim \sqrt {\alpha}\) at \(\alpha \to \infty\). Cited in 5 ReviewsCited in 34 Documents MSC: 60G40 Stopping times; optimal stopping problems; gambling theory 60J60 Diffusion processes 60G48 Generalizations of martingales Keywords:moving boundary problem for parabolic equations; local martingales; semimartingales; Dirichlet processes; local time; processes with reflection; Bessel processes; optimal stopping; maximal inequalities PDFBibTeX XMLCite \textit{L. E. Dubins} et al., Teor. Veroyatn. Primen. 38, No. 2, 288--330 (1993; Zbl 0807.60040); translation from Teor. Veroyatn. Primen. 38, No. 2, 288--330 (1993)