Tokyo Journal of Mathematics

Lévy Processes with Negative Drift Conditioned to Stay Positive

Katsuhiro HIRANO

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Abstract

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 $(-\infty,0]$.

Article information

Source
Tokyo J. of Math. Volume 24, Number 1 (2001), 291-308.

Dates
First available in Project Euclid: 19 October 2009

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1255958329

Digital Object Identifier
doi:10.3836/tjm/1255958329

Mathematical Reviews number (MathSciNet)
MR1844435

Zentralblatt MATH identifier
1020.60040

Citation

HIRANO, Katsuhiro. Lévy Processes with Negative Drift Conditioned to Stay Positive. Tokyo J. of Math. 24 (2001), no. 1, 291--308. doi:10.3836/tjm/1255958329. https://projecteuclid.org/euclid.tjm/1255958329


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