Asymptotic behaviour of first passage time distributions for Lévy processes

Details Asymptotic behaviour of first passage time distributions for Lévy processes Asymptotic behaviour of first passage time distributions for Lévy processes

2011-07-22

arxiv.org/abs/1107.4415v1

Mathematics

Probability

Scientific paper

Let $X$ be a real valued L\'evy 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 $x>D c(\cdot),$ 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.

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