Lévy Processes with Negative Drift Conditioned to Stay Positive

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 $(-\infty,0]$.