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July, 1987 Stopping Times of Bessel Processes
R. Dante DeBlassie
Ann. Probab. 15(3): 1044-1051 (July, 1987). DOI: 10.1214/aop/1176992079

Abstract

Let $X^x_\alpha$ be a Bessel process with parameter $\alpha$, starting at $x \geq 0$. Gordon [3] obtained $L^p$ inequalities which relate stopping times to stopping places for the case $\alpha = 1, x = 0$ and $p > \frac{1}{2}$. Rosenkrantz and Sawyer [5] extended them to $\alpha > 0, x = 0$ and $p \geq 1$. Burkholder [1] obtained results for $\alpha$ a positive integer, $x \geq 0$ and $p > 0$. Here we consider arbitrary starting points $x, \alpha > 0$ and $p > 0$. The $L^p$ inequalities are valid for $\alpha \geq 2$ with $p > 0$, and also for $0 < \alpha < 2$ with $p > (2 - \alpha)/2$. Examples are constructed to show that for $0 < \alpha < 2$ with $p \leq (2 - \alpha)/2$, the $L^p$ inequalities cannot hold.

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R. Dante DeBlassie. "Stopping Times of Bessel Processes." Ann. Probab. 15 (3) 1044 - 1051, July, 1987. https://doi.org/10.1214/aop/1176992079

Information

Published: July, 1987
First available in Project Euclid: 19 April 2007

zbMATH: 0645.60082
MathSciNet: MR893912
Digital Object Identifier: 10.1214/aop/1176992079

Subjects:
Primary: 60J60
Secondary: 60G40

Keywords: $L^p$-inequalities , Bessel processes , stopping times

Rights: Copyright © 1987 Institute of Mathematical Statistics

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Vol.15 • No. 3 • July, 1987
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