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- Title
Asymptotic behaviour of first passage time distributions for Lévy processes.
- Authors
Doney, R. A.; Rivero, V.
- 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 Vatutin and Wachtel (Probab Theory Relat Fields 143(1–2):177–217, 2009 ) and Doney (Probab Theory Relat Fields 152(3–4):559–588, 2012 ), 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 order 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.
- Subjects
DISTRIBUTION (Probability theory); LEVY processes; RANDOM walks; STABILITY theory; FLUCTUATIONS (Physics); PROBABILITY theory
- Publication
Probability Theory & Related Fields, 2013, Vol 157, Issue 1/2, p1
- ISSN
0178-8051
- Publication type
Article
- DOI
10.1007/s00440-012-0448-x