Parisian quasi-stationary distributions for asymmetric Lévy processes

Abstract

In recent years there has been some focus on quasi-stationary behaviour of an one-dimensional Lévy process $X$, where we ask for the law $P(X_t\in dy|{\tau }_0^{-}>t)$ for $t\to \infty$ and ${\tau }_0^{-}=\inf \{t\ge 0:X_t<0\}$. 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 $\theta$.