Open Access
January 2002 Stability of the Overshoot for Lévy Processes
R.A. Doney, R.A. Maller
Ann. Probab. 30(1): 188-212 (January 2002). DOI: 10.1214/aop/1020107765


We give equivalences for conditions like $X(T(r))/r\rightarrow 1$ and $X(T^{*}(r))/\allowbreak r\rightarrow 1$, where the convergence is in probability or almost sure, both as $r\rightarrow 0$ and $r\rightarrow \infty$, where $X$ is a L\'{e}vy process and $T(r)$ and $T^{*}(r)$ are the first exit times of $X$ out of the strip $\{(t,y):t> 0,|y|\leq r\}$ and half-plane $\{(t,y):t> 0$, $y\leq r\}$, respectively. We also show, using a result of Kesten, that $X(T^{*}(r))/r\rightarrow 1$ a.s.\ as $r\to 0$ is equivalent to $X$ ``creeping'' across a level.


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R.A. Doney. R.A. Maller. "Stability of the Overshoot for Lévy Processes." Ann. Probab. 30 (1) 188 - 212, January 2002.


Published: January 2002
First available in Project Euclid: 29 April 2002

zbMATH: 1016.60052
Digital Object Identifier: 10.1214/aop/1020107765

Primary: 60G17 , 60G51

Keywords: exit times , first passage times , local behavior , processes with independent increments

Rights: Copyright © 2002 Institute of Mathematical Statistics

Vol.30 • No. 1 • January 2002
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