The First Passage Time of a Stable Process Conditioned to Not Overshoot

Abstract

Consider a stable Lévy process $X=(X_t,t\ge 0)$ and let $T_{x}$, for $x>0$, denote the first passage time of $X$ above the level $x$. In this work, we give an alternative proof of the absolute continuity of the law of $T_{x}$ and we obtain a new expression for its density function. Our constructive approach provides a new insight into the study of the law of $T_{x}$. The random variable $T_{x}^{0}$, defined as the limit of $T_{x}$ when the corresponding overshoot tends to $0$, plays an important role in obtaining these results. Moreover, we establish a relation between the random variable $T_{x}^{0}$ and the dual process conditioned to die at $0$. This relation allows us to link the expression of the density function of the law of $T_{x}$ presented in this paper to the already known results on this topic.