On the First Passage Time and Leapover Properties of Levy Motions

TL;DR

This paper studies the first passage times and leapover distances of Levy stable random motions, revealing new insights into their crossing behavior and how far they overshoot targets.

Contribution

It introduces the first analysis of leapover properties for Levy motions, complementing existing work on first passage times.

Findings

Characterizes FPT and FPL distributions for different Levy subclasses

Provides analytical insights into overshoot behavior of Levy processes

Enhances understanding of Levy motion crossing dynamics

Abstract

We investigate two coupled properties of Levy stable random motions: The first passage times (FPTs) and the first passage leapovers (FPLs). While, in general, the FPT problem has been studied quite extensively, the FPL problem has hardly attracted any attention. Considering a particle that starts at the origin and performs random jumps with independent increments chosen from a Levy stable probability law $\lambda_(alpha,beta)(x)$, the FPT measures how long it takes the particle to arrive at or cross a target. The FPL addresses a different question: Given that the first passage jump crosses the target, then how far does it get beyond the target? These two properties are investigated for three subclasses of Levy stable motions: (i) symmetric Levy motions characterized by Levy index $\alpha$ ($0<\alpha<2$) and skewness parameter $\beta=0$, (ii) one-sided Levy motions with $0<\alpha<1$, $\beta=1$, and (iii) two-sided skewed Levy motions, the extreme case, $1<\alpha<2$, $\beta=-1$.