Mid-concavity of survival probability for isotropic Levy processes

Abstract

Let $X$ be a symmetric, pure jump, unimodal Levy process in $\mathbb{R}$ with an infinite Levy 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 $(-a,a)\times F\subset {\mathbb{R}}^d$.