First passage times of Lévy processes over a one-sided moving boundary

Details First passage times of Lévy processes over a one-sided moving boundary First passage times of Lévy processes over a one-sided moving boundary

2012-01-05

arxiv.org/abs/1201.1118v1

Mathematics

Probability

30 pages

Scientific paper

We study the one-sided exit problem with a moving boundary for L\'evy processes. Our main result states that if the boundary behaves asymptotically as $t^{\gamma}$ for some $\gamma<1/2$ then the probability that the process stays below the boundary behaves as in the case of a constant boundary. This class of boundaries is independent of Spitzer's condition in contrast to previously known results. Both positive and negative boundaries are considered. To this aim, we develop a new technique using an iteration method to reduce the exponent $\gamma$ of the boundary in each step such that the boundary eventually turns into a constant boundary. These results extend the findings of Greenwood and Novikov (1986) and are also motivated by results in the case of Brownian motion, for which the above result was proved in Uchiyama (1980).

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