## The Annals of Probability

### Stopping Times of Bessel Processes

R. Dante DeBlassie

#### 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

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