Mid-concavity of survival probability for isotropic Levy processes

TL;DR

This paper establishes the mid-concavity of survival probabilities for symmetric unimodal Levy processes in one and higher dimensions, revealing a specific concavity property of the likelihood of remaining within a set over time.

Contribution

It proves the mid-concavity of survival probabilities for a broad class of symmetric unimodal Levy processes, extending understanding of their exit time behaviors.

Findings

Survival probability is nondecreasing on (-a,0]

Survival probability is nonincreasing on [0,a)

Survival probability is concave on (-a/2,a/2)

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