The Annals of Probability

Stopping Times of Bessel Processes

R. Dante DeBlassie

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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.

Article information

Source
Ann. Probab., Volume 15, Number 3 (1987), 1044-1051.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176992079

Digital Object Identifier
doi:10.1214/aop/1176992079

Mathematical Reviews number (MathSciNet)
MR893912

Zentralblatt MATH identifier
0645.60082

JSTOR
links.jstor.org

Subjects
Primary: 60J60: Diffusion processes [See also 58J65]
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

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

Citation

DeBlassie, R. Dante. Stopping Times of Bessel Processes. Ann. Probab. 15 (1987), no. 3, 1044--1051. doi:10.1214/aop/1176992079. https://projecteuclid.org/euclid.aop/1176992079


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