Electronic Communications in Probability

Mid-concavity of survival probability for isotropic Lévy processes

Tadeusz Kulczycki

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Abstract

Let $X$ be a symmetric, pure jump, unimodal Lévy process in $\mathbb{R} $ with an infinite Lévy measure. We prove that for any fixed $t > 0$ the survival probability $P^x(\tau _{(-a,a)} > t)$ is nondecreasing on $(-a,0]$, nonincreasing on $[0,a)$ and concave on $(-a/2,a/2)$, where $a > 0$ and $\tau _{(-a,a)}$ is the first exit time of the process $X$ from $(-a,a)$. We also show a similar statement for sets $(-a,a) \times F \subset \mathbb{R} ^d$.

Article information

Source
Electron. Commun. Probab., Volume 21 (2016), paper no. 29, 9 pp.

Dates
Received: 28 September 2015
Accepted: 13 March 2016
First available in Project Euclid: 5 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1459878296

Digital Object Identifier
doi:10.1214/16-ECP4591

Mathematical Reviews number (MathSciNet)
MR3485398

Zentralblatt MATH identifier
1338.60128

Subjects
Primary: 60G51: Processes with independent increments; Lévy processes

Keywords
Lévy process exit time survival probability concavity first eigenfunction

Rights
Creative Commons Attribution 4.0 International License.

Citation

Kulczycki, Tadeusz. Mid-concavity of survival probability for isotropic Lévy processes. Electron. Commun. Probab. 21 (2016), paper no. 29, 9 pp. doi:10.1214/16-ECP4591. https://projecteuclid.org/euclid.ecp/1459878296


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