Open Access
2016 Mid-concavity of survival probability for isotropic Lévy processes
Tadeusz Kulczycki
Electron. Commun. Probab. 21: 1-9 (2016). DOI: 10.1214/16-ECP4591


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 $(-a,a) \times F \subset \mathbb{R} ^d$.


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Tadeusz Kulczycki. "Mid-concavity of survival probability for isotropic Lévy processes." Electron. Commun. Probab. 21 1 - 9, 2016.


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

zbMATH: 1338.60128
MathSciNet: MR3485398
Digital Object Identifier: 10.1214/16-ECP4591

Primary: 60G51

Keywords: concavity , Exit time , first eigenfunction , Lévy process , Survival probability

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