Compactness and continuity properties for a Lévy process at a two-sided exit time

Abstract

We consider a Lévy process $X=(X(t))_{t\ge 0}$ in a generalised Feller class at 0, and study the exit position, $\left \vert X(T(r))\right \vert $, as $X$ leaves, and the position, $\left \vert X(T( r) -)\right \vert $, just prior to its leaving, at time $T(r)$, a two-sided region with boundaries at $\pm r$, $r>0$. Conditions are known for $X$ to be in the Feller class $FC_{0}$ at zero, by which we mean that each sequence $t_{k}\downarrow 0$ contains a subsequence through which $X(t_{k})$, after norming by a nonstochastic function, converges to an a.s. finite nondegenerate random variable. We use these conditions on $X$ to characterise similar properties for the normed positions $\left \vert X(T( r))\right \vert /r$ and $\left \vert X(T( r) -)\right \vert /r$, and also for the normed jump $\left \vert \Delta X(T(r))/r\right \vert = \left \vert X(T(r))-X(T(r)-)\right \vert /r$ (“the jump causing ruin"), as convergence takes place through sequences $r_{k}\downarrow 0$. We go on to give conditions for the continuity of distributions of the limiting random variables obtained in this way.