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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 \(\alpha \to \infty\).

MSC:

60G40 Stopping times; optimal stopping problems; gambling theory
60J60 Diffusion processes
60G48 Generalizations of martingales
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