Philip S. Griffin and Ross A. Maller
Abstract: This paper is concerned with the small time behaviour of a Lévy process $X$. In particular, we investigate the stabilities of the times at which $X$, started with $X_0=0$, first leaves the space-time regions $\{(t,y)\in\mathbb R^2: y\le rt^b, t\ge 0\}$ (one-sided exit), or $\{(t,y)\in\mathbb R^2: |y|\le rt^b, t\ge 0\}$ (two-sided exit), $0\le b<1$, as $r\searrow 0$. 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.
Journal: Ann. Inst. Henri Poincaré Probab. Stat. 49 (2013), no. 1, 208–235
DOI: 10.1214/11-AIHP449