Parisian quasi-stationary distributions for asymmetric L\'evy processes

Abstract

In recent years there has been some focus on quasi-stationary behaviour of an one-dimensional L\'evy process $X$, where we ask for the law $P(X_t\in dy | \tau^-_0>t)$ for $t\to\infty$ and $\tau_0^-=\inf\{t\geq 0: X_t<0\}$. In this paper we address the same question for so-called Parisian ruin time $\tau^\theta$, that happens when process stays below zero longer than independent exponential random variable with intensity $\theta$.