Small and Large Time Stability of the Time taken for a L\'evy Process to Cross Curved Boundaries

TL;DR

This paper studies the small and large time stability of the first exit times of a Lévy process crossing curved boundaries, analyzing their convergence modes and relation to the process's stability.

Contribution

It characterizes the stability modes of exit times for Lévy processes crossing curved boundaries at small and large times, linking them to the process's stability properties.

Findings

Exit times behave like deterministic functions under certain stability conditions.

Convergence modes include probability, almost surely, and in L^p.

Stability of exit times is often equivalent to the process's relative stability.

Abstract

This paper is concerned with the small time behaviour of a L\'{e}vy process $X$. In particular, we investigate the {\it stabilities} of the times, $\Tstarb(r)$ and $\Tbarb(r)$, at which $X$, started with $X_0=0$, first leaves the space-time regions $\{(t,y)\in\R^2: y\le rt^b, t\ge 0\}$ (one-sided exit), or $\{(t,y)\in\R^2: |y|\le rt^b, t\ge 0\}$ (two-sided exit), $0\le b<1$, as $r\dto 0$. Thus essentially we determine whether or not these passage times behave like deterministic functions in the sense of different modes of convergence; specifically convergence in probability, almost surely and in $L^p$. In many instances these are seen to be equivalent to relative stability of the process $X$ itself. The analogous large time problem is also discussed.