## The Annals of Probability

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

### Stopping Times of Bessel Processes

#### 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