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June 2001 Lévy Processes with Negative Drift Conditioned to Stay Positive
Katsuhiro HIRANO
Tokyo J. Math. 24(1): 291-308 (June 2001). DOI: 10.3836/tjm/1255958329


Let $X$ be a Lévy process with negative drift starting from $x>0$, and let $\tau$ and $\tau_s$ be the first passage times to $(-\infty,0]$ and $(s,\infty)$, respectively. Under appropriate exponential moment conditions of $X$, we show that, for every $A\in\mathcal{F}_t$, the conditional laws $P_x(X\in A | \tau>s)$ and $P_x(X\in A | \tau>\tau_s)$ converge to different distributions as $s\rightarrow\infty$. Both of them can be regarded as the laws of $X$ conditioned to stay positive. We characterize these limit laws in terms of $h$-transforms, by the renewal functions, of some Lévy processes killed at the entrance time into $(-\infty,0]$.


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Katsuhiro HIRANO. "Lévy Processes with Negative Drift Conditioned to Stay Positive." Tokyo J. Math. 24 (1) 291 - 308, June 2001.


Published: June 2001
First available in Project Euclid: 19 October 2009

zbMATH: 1020.60040
MathSciNet: MR1844435
Digital Object Identifier: 10.3836/tjm/1255958329

Rights: Copyright © 2001 Publication Committee for the Tokyo Journal of Mathematics

Vol.24 • No. 1 • June 2001
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