ScholarlyCommons

Title

Optimal Stopping Rules and Maximal Inequalities for Bessel Processes

Author(s)

L. E. Dubins
Larry A. Shepp, University of Pennsylvania
A. N. Shiryaev

Document Type

Journal Article

Date of this Version

1994

Publication Source

Theory of Probability & Its Applications

Volume

38

Issue

2

Start Page

226

Last Page

261

DOI

10.1137/1138024

Abstract

We consider, for Bessel processes X ∈ Bes^α 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 X_rr≤r ≤ γ(α) is a constant depending on the dimension (order) α. It is shown that γ(α) ∼ √α at α → ∞.

Copyright/Permission Statement

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

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

Recommended Citation

Dubins, L. E., Shepp, L. A., & Shiryaev, A. N. (1994). Optimal Stopping Rules and Maximal Inequalities for Bessel Processes. Theory of Probability & Its Applications, 38 (2), 226-261. http://dx.doi.org/10.1137/1138024