Optimal Stopping Rules and Maximal Inequalities for Bessel Processes
Lester E. Dubins, Larry A Shepp, Albert Nikolaevich Shiryaev · Theory of Probability and Its Applications · 1994
We consider, for Bessel processes $X \in {\operatorname{Bes}}^\alpha (x)$ with arbitrary order (dimension) $\alpha \in {\bf R}$, 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 \[ {\bf E}\mathop {\max }\limits_{r \leq \tau} X_r \leq \gamma (\alpha )\sqrt {{\bf E}\tau }, \] 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 $.