## Tokyo Journal of Mathematics

### Lévy Processes with Negative Drift Conditioned to Stay Positive

Katsuhiro HIRANO

#### 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