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Optimal stopping rules and maximal inequalities for Bessel processes

L. E. Dubins^a, L. A. Shepp^b, A. N. Shiryaev<sup>c

a</sup> Department of Mathematics, University of California, Berkeley, CA, USA
^b AT&T Bell Laboratories, Murray Hill, New Jersey, USA
^c Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: We consider, for Bessel processes $X\in\operatorname{Bes}^\alpha(x)$ with arbitrary order (dimension) $\alpha \in \mathbf{R}$, 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
$$ \mathbf{E}\max\limits_{r\le\tau}X_r\le\gamma(\alpha)\sqrt {\mathbf{E}\tau}, $$
where $X \in\operatorname{Bes}^\alpha(0)$, $\tau$ 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 $\alpha\to\infty$.

Keywords: 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.

Received: 02.10.1992

 English version: Theory of Probability and its Applications, 1993, 38:2, 226–261

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