Asymptotic behaviour of first passage time distributions for Lévy processes

Abstract

Let $X$ be a real valued Lévy process that is in the domain of attraction of a stable law without centering with norming function $c.$ As an analogue of the random walk results in \cite{vw} and \cite{rad} we study the local behaviour of the distribution of the lifetime $\zeta$ under the characteristic measure $\underline{n}$ of excursions away from 0 of the process $X$ reflected in its past infimum, and of the first passage time of $X$ below $0,$ $T_0=\inf \{t>0:X_t<0\},$ under ${\mathbb{P}}_x(\cdot ),$ for $x>0,$ in two different regimes for $x,$ viz. $x=o(c(\cdot ))$ and $x>Dc(\cdot ),$ for some $D>0.$ We sharpen our estimates by distinguishing between two types of path behaviour, viz. continuous passage at $T_0$ and discontinuous passage. In the way to prove our main results we establish some sharp local estimates for the entrance law of the excursion process associated to $X$ reflected in its past infimum.

References in corpus (1)

• Recurrent extensions of self-similar Markov processes and Cramér's condition II