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

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, $\text{Tstarb}(r)$ and $\text{Tbarb}(r)$, at which $X$, started with $X_0=0$, first leaves the space-time regions $\{(t,y)\in {\mathbb{R}}^2:y\le r{t}^b,t\ge 0\}$ (one-sided exit), or $\{(t,y)\in {\mathbb{R}}^2:\mathrm{\mid }y\mathrm{\mid }\le r{t}^b,t\ge 0\}$ (two-sided exit), $0\le b<1$, as $r\text{dto}0$. 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.